These are my live-TeXed notes for the course Math 223a: Algebraic Number Theory taught by Joe Rabinoff at Harvard, Fall 2012.
09/05/2012
Introduction
This is a one-year course on class field theory — one huge piece of intellectual work in the 20th century.
Recall that a global field is either a finite extension of
(characteristic 0) or a field of rational functions on a projective curve over a field of characteristic
(i.e., finite extensions of
). A local field is either a finite extension of
(characteristic 0) or a finite extension of
(and sometimes we also include
and
as local fields) . The major goal of class field theory is to describe all abelian extensions of local and global fields
(an abelian extension means a Galois extension with an abelian Galois group). Suppose
is the maximal abelian extension of
, then
, the topological abelianization of the absolute Galois group
. Moreover, there is a bijection between abelian extensions of
and closed subgroups of
. So we would like to understand the structure of
.
We also would like to know information about ramification of abelian extensions. For example, does
have a degree 3 extension ramified only over 5? This can be nicely answered by class field theory. Class field theory also allows us to classify infinite abelian extensions via studying the topological group
. The course will start with lots of topological groups in the first week and one may be impressed by how seemingly unrelated to number theory at first glimpse.
Here some useful applications of class field theory.
(Primes in arithmetic progressions)
The famous Dirichlet theorem says that for an integer
![$m\ge2$](./latex/latex2png-ClassFieldTheory_50624157_-3.gif)
, the primes
![$p\nmid m$](./latex/latex2png-ClassFieldTheory_242242747_-5.gif)
is equidistributed in
![$(\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_203733105_-5.gif)
. Notice that there is a canonical isomorphism between
![$\Gal(\mathbb{Q}(\zeta_m)/\mathbb{Q})$](./latex/latex2png-ClassFieldTheory_69217629_-5.gif)
and
![$(\mathbb{Z}/ m \mathbb{Z} )^\times$](./latex/latex2png-ClassFieldTheory_260995917_-5.gif)
and this isomorphism sends the Frobenius element associated to any
![$p\nmid m$](./latex/latex2png-ClassFieldTheory_242242747_-5.gif)
to
![$p\bmod{ m}$](./latex/latex2png-ClassFieldTheory_41460285_-4.gif)
. So the classical theorem of Dirichlet can be viewed as a special case of the following Chebotarev's density theorem.
Let
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
be a finite Galois extension. Then the Frobenius elements (conjugacy classes)
![$\mathrm{Frob}_\mathfrak{p}$](./latex/latex2png-ClassFieldTheory_40847091_-5.gif)
for primes
![$\mathfrak{p} $](./latex/latex2png-ClassFieldTheory_192324747_-4.gif)
of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
are equidistributed on the conjugate classes of
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
.
Chebotarev's density theorem is proved via reducing to the case of cyclic extensions using a nice counting argument and then applying class field theory (cf. 34).
(Artin
-functions)
An
Artin representation is a continuous representation
![$\rho: G_K\rightarrow GL(V)$](./latex/latex2png-ClassFieldTheory_161047895_-5.gif)
where
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field and
![$V$](./latex/latex2png-ClassFieldTheory_43545618_-1.gif)
is a finite complex vector space. One can attach an
Artin
-function to each Artin representation. When
![$V$](./latex/latex2png-ClassFieldTheory_43545618_-1.gif)
is one-dimensional, an Artin representation is simply a character of
![$G_K\rightarrow \mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_11237339_-2.gif)
, which must factor through
![$G_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_227018301_-5.gif)
. So a one-dimensional Artin representation is nothing but a character of
![$G_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_227018301_-5.gif)
. By continuity, this character factors through
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
, where
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
is a finite abelian extension of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
Weber generalized the Dirichlet
-functions to Weber
-functions over any number fields. He proved that a Weber
-function has analytic continuation to the whole
and satisfies a functional equation. Class field theory then tells us that the Weber
-functions are exactly the one-dimensional Artin
-functions.
Let
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
be a smooth projective connected curve over a finite field and
![$K=K(X)$](./latex/latex2png-ClassFieldTheory_39084797_-5.gif)
be the function field of
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
. Then class field theory classifies all abelian covers of
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
. In particular, this gives the abelianization of the etale fundamental group
![$\pi_1(X)$](./latex/latex2png-ClassFieldTheory_36339580_-5.gif)
of
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
. The proof of Weil conjecture II uses it in an essential way.
(Cohomology of
)
In the second semester we will study the Tate global and local duality, Brauer groups and introduce all the cohomological machinery in order to prove class field theory.
Now we briefly turn to the main statements of class field theory. Class field theory gives Artin maps
(in the global case) and
and the kernel and image of the Artin maps can be described. The crucial thing is that the source of the Artin maps are intrinsic to the field
(doesn't involve
). Moreover the Artin maps satisfy the local-global compatibility: the diagram
commutes. In other words, class field theory is functorial in
. For a finite abelian extension
, the Artin map induces the relative Artin maps
and
. They are surjective and the kernel is exactly the norm subgroups. We can furthermore read ramification data from the relative Artin maps. In the local case
is exactly the inertia group
and
is exactly the
-th ramification group of
. In the global case, the ramification data can be extracted from the local-global compatibility. We will spend most of the first semester to state the class field theory and draw important consequences from it and devote the second semester to the proofs.
Here are a few words about the proofs of class field theory. The classical approach is to do the global case first, using cyclotomic extensions, Kummer extensions and Artin-Schreier extensions (in characteristic
) to fill up the absolute Galois group, and then derive the local case from it. The cohomological approach is to establish local class field theory using group cohomology and then "glue" the local Artin maps to obtain the global Artin maps. One of the advantage of the cohomological approach is that the local-global compatibility comes from the construction. We will take this approach in the second semester.
Finally we may also talk about explicit class field theory, i.e., finding explicit construction (e.g. as splitting field of polynomials) of abelian extensions. This is a highly open problem in general with several known cases:
- When
,
, the union of all cyclotomic extensions of
. So the polynomials
exhaust all abelian extensions of
. This the most satisfactory case.
- When
is an imaginary quadratic field. The CM theory of elliptic curves assert that
can be obtained essentially by adjoining all the torsion points on an elliptic curve with complex multiplication by
.
- When
is a global function field, there is a theory of Drinfeld modules to obtain most abelian extensions of
(apart from some ramification restriction).
- When
is a nonarchimedean local field, Lubin-Tate theory tells that
can be obtained by adjoining all torsion points of the Lubin-Tate formal groups.
Somehow adjoining torsion points of a group law is possibly the only known way to construct explicit class fields.
09/07/2012
Global fields
Today and next Monday we will review the basic notions we learned from Math 129, taking this opportunity to set up the notations.
A
number field is a finite extension of
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
. It is an abstract field and have many embeddings into the complex numbers
![$\mathbb{C}$](./latex/latex2png-ClassFieldTheory_26235984_-1.gif)
(we do not specify one).
A
global function field is a finite (separable) extension of
![$\mathbb{F}_q(t)$](./latex/latex2png-ClassFieldTheory_13471827_-5.gif)
. It is a fact that the algebraic closure
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
of
![$\mathbb{F}_p$](./latex/latex2png-ClassFieldTheory_68858419_-5.gif)
in a global function field
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a finite field, called the
field of constants. Equivalently,
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is the field of rational functions on a smooth projective geometrically connected curve
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
over
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
, unique up to isomorphism. The geometrically connectedness ensures that
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
is the field of constants.
A global field is either a number field or a global function field.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field. A
place of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is an equivalent class
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
of nontrivial absolute values
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
on
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. Two absolute values
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
and
![$|\cdot|'$](./latex/latex2png-ClassFieldTheory_26416130_-5.gif)
are equivalent if and only if they induce the same topology on
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
under the metric
![$d(x,y)=|x-y|$](./latex/latex2png-ClassFieldTheory_6835869_-5.gif)
, if and only if
![$|\cdot|'{}=|\cdot|^\alpha$](./latex/latex2png-ClassFieldTheory_161502789_-5.gif)
for
![$\alpha\in \mathbb{R}_{>0}$](./latex/latex2png-ClassFieldTheory_137311512_-4.gif)
. The set of all places of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is denoted by
![$V_K$](./latex/latex2png-ClassFieldTheory_63081452_-2.gif)
. Suppose
![$v\in V_K$](./latex/latex2png-ClassFieldTheory_60378006_-2.gif)
, then we have the
completion ![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
with respect to any absolute values
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
corresponding to
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
.
When
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
, there is only one archimedean place
![$\infty$](./latex/latex2png-ClassFieldTheory_51617303_0.gif)
given by the usual absolute value and
![$\mathbb{Q}_\infty=\mathbb{R}$](./latex/latex2png-ClassFieldTheory_5182301_-3.gif)
. For any prime number
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
, the
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
-adic absolute value
![$||\cdot||_p$](./latex/latex2png-ClassFieldTheory_203791146_-5.gif)
is defined on the generators of
![$\mathbb{Q}^\times\cong \{\pm1\}\times\bigoplus_p p^\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_17355450_-8.gif)
by
![$$||p||_p=\frac{1}{p},\quad ||\ell||_p=1,\quad ||-1||_p=1.$$](./latex/latex2png-ClassFieldTheory_742369_.gif)
In this way we have a bijection between the set of primes of
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
and all non-archimedean places of
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
. The completion with respect to
![$||\cdot||_p$](./latex/latex2png-ClassFieldTheory_203791146_-5.gif)
is denoted by
![$\mathbb{Q}_p$](./latex/latex2png-ClassFieldTheory_199577036_-5.gif)
.
Suppose
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is an archimedean place of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
, then there exists an absolute value
![$|\cdot|_v$](./latex/latex2png-ClassFieldTheory_52792546_-5.gif)
such that
![$|n|_v=n$](./latex/latex2png-ClassFieldTheory_35327180_-5.gif)
for any
![$n\in \mathbb{Z}_{n\ge0}$](./latex/latex2png-ClassFieldTheory_257828885_-6.gif)
. By the Gelfand-Mazur theorem,
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
is either
![$\mathbb{R}$](./latex/latex2png-ClassFieldTheory_41964624_0.gif)
or
![$\mathbb{C}$](./latex/latex2png-ClassFieldTheory_26235984_-1.gif)
, so the absolute value
![$|\cdot|_v$](./latex/latex2png-ClassFieldTheory_52792546_-5.gif)
is the usual absolute value on
![$\mathbb{R}$](./latex/latex2png-ClassFieldTheory_41964624_0.gif)
or
![$\mathbb{C}$](./latex/latex2png-ClassFieldTheory_26235984_-1.gif)
. The
normalized absolute value is often defined as
![$||\cdot||_v=|\cdot|_v$](./latex/latex2png-ClassFieldTheory_61130927_-5.gif)
(when
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is real) and
![$||\cdot||_v=|\cdot|_v^2$](./latex/latex2png-ClassFieldTheory_19418098_-5.gif)
(when
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is complex). This normalization simplifies many statements like the product formula.
Suppose
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is a non-archimedean place of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. Let
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
be a representative of
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
. Then
![$-\log|K^\times|\subseteq \mathbb{R}$](./latex/latex2png-ClassFieldTheory_224555793_-5.gif)
is discrete (Ostrowski's theorem), hence is equal to
![$\alpha \mathbb{Z}$](./latex/latex2png-ClassFieldTheory_241160941_0.gif)
for some
![$\alpha\in \mathbb{R}_{>0}$](./latex/latex2png-ClassFieldTheory_137311512_-4.gif)
. We normalized the valuation
![$\ord_v:K\rightarrow \mathbb{Z}\cup\{\infty\}$](./latex/latex2png-ClassFieldTheory_109768817_-5.gif)
such that
![$\ord_v(x)=-\frac{1}{\alpha}\log|x|$](./latex/latex2png-ClassFieldTheory_47702025_-6.gif)
. This valuation is intrinsic to
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
and is called the
normalized valuation. It extends uniquely to
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
.
We denote
![$\mathcal{O}_v=\{x\in K_v: \ord_v(x)\ge0\}$](./latex/latex2png-ClassFieldTheory_7522_-5.gif)
, the
valuation ring of
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
;
![$\mathfrak{m}_v=\{x\in \mathcal{O}_v: \ord_v(x)>0\}$](./latex/latex2png-ClassFieldTheory_85561980_-5.gif)
, the
maximal ideal of
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
and
![$k_v=\mathcal{O}_v/\mathfrak{m}_v$](./latex/latex2png-ClassFieldTheory_113231457_-5.gif)
, the
residue field of
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
(a finite field). We define the
normalized
-adic absolute value to be
![$||x||_v=(\# k_v)^{-\ord_v(x)}$](./latex/latex2png-ClassFieldTheory_254354467_-5.gif)
. It is equivalent to absolute value we started with.
(Product formula)
For
![$x\in K^\times$](./latex/latex2png-ClassFieldTheory_22902088_-1.gif)
,
![$||x||_v=1$](./latex/latex2png-ClassFieldTheory_77400492_-5.gif)
for almost all places
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
and
Let
![$S$](./latex/latex2png-ClassFieldTheory_43349010_-1.gif)
be a nonempty finite set of places of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
containing the set
![$S_\infty$](./latex/latex2png-ClassFieldTheory_54331671_-2.gif)
of archimedean places. The
ring of
-integer is
![$\mathcal{O}_{K,S}=\{x\in K: ||x||_v\le1,\forall v\not\in S\}$](./latex/latex2png-ClassFieldTheory_264985633_-5.gif)
. This is a Dedekind domain. For
![$v\not\in S$](./latex/latex2png-ClassFieldTheory_26615858_-4.gif)
, we denote
![$\mathfrak{p}_v=\{x\in \mathcal{O}_{K,s}: ||x||_v<1\}=\mathcal{O}_{K,S}\cap \mathfrak{m}_v$](./latex/latex2png-ClassFieldTheory_39155552_-5.gif)
,
![$\mathcal{O}_v=\varprojlim_n \mathcal{O}_{K,S}/\mathfrak{p}_v^n$](./latex/latex2png-ClassFieldTheory_36779178_-10.gif)
and
![$k_v=\mathcal{O}_v/\mathfrak{p}_v$](./latex/latex2png-ClassFieldTheory_155203999_-5.gif)
.
Every maximal ideal of
![$\mathcal{O}_{K,S}$](./latex/latex2png-ClassFieldTheory_239055523_-5.gif)
is of the form
![$\mathfrak{p}_v$](./latex/latex2png-ClassFieldTheory_75269214_-4.gif)
for
![$v\not\in S$](./latex/latex2png-ClassFieldTheory_26615858_-4.gif)
.
For
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
,
![$\mathcal{O}_{K,S}=\mathbb{Z}[\frac{1}{p}, p\in S]$](./latex/latex2png-ClassFieldTheory_74685940_-9.gif)
. For any prime
![$p\not\in S$](./latex/latex2png-ClassFieldTheory_20324402_-4.gif)
,
![$\mathcal{O}_p\cong \mathbb{Z}_p$](./latex/latex2png-ClassFieldTheory_101360890_-5.gif)
.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a number field.
![$\mathcal{O}_K=\mathcal{O}_{K,S_\infty}$](./latex/latex2png-ClassFieldTheory_42751374_-5.gif)
is called the
ring of integers of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. It is the integral closure of
![$\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_50353232_0.gif)
in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
Let
![$K=\mathbb{F}_q(t)$](./latex/latex2png-ClassFieldTheory_56238163_-5.gif)
. Then taking the usual degree of rational function gives exactly the valuation associated to the closed point
![$\infty\in \mathbb{P}^1$](./latex/latex2png-ClassFieldTheory_118153622_-1.gif)
. Moreover,
![$\mathcal{O}_{\mathbb{F}_q(t),\infty}=\mathbb{F}_q[t]$](./latex/latex2png-ClassFieldTheory_204960781_-7.gif)
. A homomorphism
![$\mathbb{F}_q(t)\rightarrow K$](./latex/latex2png-ClassFieldTheory_178159379_-5.gif)
gives a map
![$\phi: X\rightarrow \mathbb{P}^1$](./latex/latex2png-ClassFieldTheory_3362104_-4.gif)
. We know that
![$S_\infty=\phi^{-1}(\infty)$](./latex/latex2png-ClassFieldTheory_261546366_-5.gif)
and
![$\mathcal{O}_{K,S_\infty}$](./latex/latex2png-ClassFieldTheory_243552768_-5.gif)
is the integral closure of
![$\mathbb{F}_q[t]$](./latex/latex2png-ClassFieldTheory_201224109_-5.gif)
in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
09/10/2012
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field and
![$S$](./latex/latex2png-ClassFieldTheory_43349010_-1.gif)
be a finite set of places of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
containing
![$S_\infty$](./latex/latex2png-ClassFieldTheory_54331671_-2.gif)
. Then
![$\mathcal{O}_{K,S}$](./latex/latex2png-ClassFieldTheory_239055523_-5.gif)
is a Dedekind domain. We denote the group of fractional ideals of
![$\mathcal{O}_{K,S}$](./latex/latex2png-ClassFieldTheory_239055523_-5.gif)
by
![$I_{K,S}$](./latex/latex2png-ClassFieldTheory_25822041_-5.gif)
. Then
![$I_{K,S}\cong\bigoplus_{\mathfrak{p}\not\in S} \mathfrak{p}^\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_44701508_-8.gif)
. Any
![$\alpha\in K^\times$](./latex/latex2png-ClassFieldTheory_144708705_-1.gif)
generates the principal fractional ideal
![$(\alpha)=\prod_{\mathfrak{p}_v}\mathfrak{p}_v^{\ord_v(\alpha)}$](./latex/latex2png-ClassFieldTheory_77359607_-8.gif)
. Let
![$P_{K,S}\subseteq I_{K,S}$](./latex/latex2png-ClassFieldTheory_92460857_-5.gif)
be the subgroup of principal ideals. We define the
-class group of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
to be
![$\Cl_S(K)=I_{K,S}/P_{K,S}$](./latex/latex2png-ClassFieldTheory_1499462_-5.gif)
.
Extensions of global fields
Let
be a finite extension of global fields and
be a finite set of places of
containing
. Let
be the preimage of
. Then natural map
makes
a finite projective
-module of rank
.
Fix a place
, we factorize
. Then the
's correspond to the places
of
restricting to
.
Denote
![$g=g(v)=g(L/v)$](./latex/latex2png-ClassFieldTheory_98322354_-5.gif)
. The number
![$e_i$](./latex/latex2png-ClassFieldTheory_53703700_-2.gif)
is called the
ramification index of
![$w_i/v$](./latex/latex2png-ClassFieldTheory_197101913_-5.gif)
, denoted by
![$e(w_i/v)=e(w_i/K)=e(\mathfrak{q}_i)$](./latex/latex2png-ClassFieldTheory_88847525_-5.gif)
. The number
![$f_i=[k_{w_i}: k_v]=f(w_i/v)$](./latex/latex2png-ClassFieldTheory_242729659_-5.gif)
is called the
residue degree of
![$w_i/v$](./latex/latex2png-ClassFieldTheory_197101913_-5.gif)
. We have
![$n=[L:K]=\sum_{i=1}^g e_if_i$](./latex/latex2png-ClassFieldTheory_166249522_-5.gif)
.
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is called
unramified at
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
if
![$e_i=1$](./latex/latex2png-ClassFieldTheory_184322411_-2.gif)
for all
![$i$](./latex/latex2png-ClassFieldTheory_42169362_0.gif)
's,
totally split at
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
if
![$g=n$](./latex/latex2png-ClassFieldTheory_51934228_-4.gif)
and
inert at
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
if
![$g=1$](./latex/latex2png-ClassFieldTheory_50557972_-4.gif)
and
![$e_1=1$](./latex/latex2png-ClassFieldTheory_84113045_-2.gif)
.
Now fix an archimedean place
. Then
is either 1 or 2.
![$w_i/v$](./latex/latex2png-ClassFieldTheory_197101913_-5.gif)
is called
unramified if
![$[L_{w_i}: K_v]=1$](./latex/latex2png-ClassFieldTheory_191480978_-5.gif)
and
ramified otherwise. Similarly we have
![$n=\sum_{i=1}^g[L_{w_i}:K_v]$](./latex/latex2png-ClassFieldTheory_134455640_-5.gif)
.
Suppose
is Galois with
, then
acts on
given by
. This action is transitive on the fibers of
.
The
decomposition group ![$D(w)=D(w/v)=D(w/K)$](./latex/latex2png-ClassFieldTheory_24550762_-5.gif)
of
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
is defined to be the stabilizer of
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
in
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
. We have
![$gD(w)g^{-1}=D(gw)$](./latex/latex2png-ClassFieldTheory_100709818_-5.gif)
. In particular, when
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is abelian the
![$D(w)$](./latex/latex2png-ClassFieldTheory_219903034_-5.gif)
's for all
![$w|v$](./latex/latex2png-ClassFieldTheory_191859692_-5.gif)
coincide, we simply denoted it by
![$D(v)$](./latex/latex2png-ClassFieldTheory_218854458_-5.gif)
.
When
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is archimedean and
![$w/v$](./latex/latex2png-ClassFieldTheory_230657004_-5.gif)
is ramified, we denote
![$\Gal(L_w/K_v)=\{1, \sigma_{w/v}\}$](./latex/latex2png-ClassFieldTheory_114254783_-6.gif)
, where
![$\sigma_{w/v}$](./latex/latex2png-ClassFieldTheory_208993404_-6.gif)
is the complex conjugation. These complex conjugations are related by
![$g\sigma_{w/v}g^{-1}=\sigma_{gw/ v}$](./latex/latex2png-ClassFieldTheory_73365711_-6.gif)
.
When
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is non-archimedean,
![$D(w)$](./latex/latex2png-ClassFieldTheory_219903034_-5.gif)
acts on the residue extension
![$k_w/k_v$](./latex/latex2png-ClassFieldTheory_113050284_-5.gif)
and gives a map
![$D(w)\rightarrow\Gal(k_w/k_v)$](./latex/latex2png-ClassFieldTheory_25820415_-5.gif)
. This map is surjective and the kernel is called the
inertia subgroup, denoted by
![$I(w)=I(w/v)=I(w/K)$](./latex/latex2png-ClassFieldTheory_19307797_-5.gif)
. Similarly we have
![$gI(w)g^{-1}=I(gw)$](./latex/latex2png-ClassFieldTheory_100689333_-5.gif)
. Counting shows that
![$\#I(w)=e$](./latex/latex2png-ClassFieldTheory_46572593_-5.gif)
. So
![$w/v$](./latex/latex2png-ClassFieldTheory_230657004_-5.gif)
is unramified if and only if
![$I(w)=\{1\}$](./latex/latex2png-ClassFieldTheory_265589059_-5.gif)
, if and only if
![$D(w)\cong\Gal(k_w/k_v)$](./latex/latex2png-ClassFieldTheory_7858931_-5.gif)
.
Suppose
![$w/v$](./latex/latex2png-ClassFieldTheory_230657004_-5.gif)
is unramified. The generator
![$x\mapsto x^{\#k_v}$](./latex/latex2png-ClassFieldTheory_116275913_-1.gif)
of the cyclic group
![$D(w)\cong\Gal(k_w/k_v)$](./latex/latex2png-ClassFieldTheory_7858931_-5.gif)
is called the
Frobenius, denoted by
![$\Frob_{w/v}$](./latex/latex2png-ClassFieldTheory_82191452_-6.gif)
(you may think it as the analogue of the complex conjugation
![$\sigma_{w/v}$](./latex/latex2png-ClassFieldTheory_208993404_-6.gif)
. Similarly we have
![$g\Frob_{w/v}g^{-1}=\Frob_{gw}$](./latex/latex2png-ClassFieldTheory_170746844_-6.gif)
. In particular, when
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is abelian we have a unique Frobenius attached to
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
, denoted by
![$\Frob_v$](./latex/latex2png-ClassFieldTheory_264039232_-2.gif)
.
Valued fields
A
valued field is a pair
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
, where
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a field and
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
is a nontrivial absolute value on
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We endow
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
with the metric
![$d(x,y)=|x-y|$](./latex/latex2png-ClassFieldTheory_6835869_-5.gif)
. We say that
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
is
complete if
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is complete under this metric.
A
valued field extension is a map
![$(K,|\cdot|)\rightarrow (L,|\cdot|')$](./latex/latex2png-ClassFieldTheory_65342839_-5.gif)
, i.e., a homomorphism
![$K\rightarrow L$](./latex/latex2png-ClassFieldTheory_109288209_-1.gif)
such that
![$|\cdot|'|_K=|\cdot|$](./latex/latex2png-ClassFieldTheory_262886870_-5.gif)
.
(Gelfand-Mazur)
If
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
is a complete archimedean field. Then
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is isomorphic to either
![$\mathbb{R}$](./latex/latex2png-ClassFieldTheory_41964624_0.gif)
or
![$\mathbb{C}$](./latex/latex2png-ClassFieldTheory_26235984_-1.gif)
.
Let
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
be a non-archimedean valued field. The ring
![$\mathcal{O}_K=\{x\in K: |x|\le1\}$](./latex/latex2png-ClassFieldTheory_181424926_-5.gif)
is called the
ring of integers. It is a valuation ring with valuation group contained in
![$\mathbb{R}$](./latex/latex2png-ClassFieldTheory_41964624_0.gif)
. Denote its fraction field by
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We similarly define the
maximal ideal ![$\mathfrak{m}_K$](./latex/latex2png-ClassFieldTheory_192969634_-2.gif)
and the
residue field ![$k=\mathcal{O}_K/\mathfrak{m}_K$](./latex/latex2png-ClassFieldTheory_116089361_-5.gif)
.
We say
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
is
discretely valued if
![$|K^\times|\subseteq \mathbb{R}_{>0}$](./latex/latex2png-ClassFieldTheory_252145055_-5.gif)
is discrete (equivalently, equal to
![$c^\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_218125432_0.gif)
for some
![$c\in(0,1)$](./latex/latex2png-ClassFieldTheory_239306339_-5.gif)
). Any element
![$\pi\in K^\times$](./latex/latex2png-ClassFieldTheory_22730519_-1.gif)
such that
![$|\pi|=c$](./latex/latex2png-ClassFieldTheory_135269918_-5.gif)
is called a
uniformizer. Let
![$\ord_K: K\rightarrow \mathbb{Z}\cup\{\infty\}$](./latex/latex2png-ClassFieldTheory_196525158_-5.gif)
be the unique valuation such that
![$\ord_K(\pi)=1$](./latex/latex2png-ClassFieldTheory_81308482_-5.gif)
.
The famous Hensel's lemma holds for any valued fields (but the proof in this generality is different from the discrete valued case).
(Hensel's lemma)
Suppose
![$f\in \mathcal{O}_K[x]$](./latex/latex2png-ClassFieldTheory_188808150_-5.gif)
is monic and
![$\alpha_1\in \mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_240227022_-2.gif)
such that
![$|f(\alpha_1)|<1$](./latex/latex2png-ClassFieldTheory_92425027_-5.gif)
and
![$|f'(\alpha_1)|=1$](./latex/latex2png-ClassFieldTheory_96023507_-5.gif)
. Then there exists
![$\alpha\in \mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_46900765_-2.gif)
such that
![$|\alpha-\alpha_1|<1$](./latex/latex2png-ClassFieldTheory_238435261_-5.gif)
and
![$f(\alpha)=0$](./latex/latex2png-ClassFieldTheory_35500840_-5.gif)
.
09/12/2012
Let
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
be a complete discretely valued field and
![$(L,|\cdot|)$](./latex/latex2png-ClassFieldTheory_240980997_-5.gif)
be a complete discretely valued field extension of
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
. Let
![$\pi_K$](./latex/latex2png-ClassFieldTheory_121468594_-2.gif)
be a uniformizer of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We define
![$e=e(L/K)=\ord_L(\pi_K)$](./latex/latex2png-ClassFieldTheory_250832736_-5.gif)
to be the
ramification index of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
and
![$f=[l: k]$](./latex/latex2png-ClassFieldTheory_79140617_-5.gif)
the
residue degree. We say that
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is
unramified if
![$e=1$](./latex/latex2png-ClassFieldTheory_17003540_0.gif)
and
![$l/k$](./latex/latex2png-ClassFieldTheory_120943636_-5.gif)
is separable (so
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is separable), or equivalently,
![$\mathcal{O}_L/\mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_233134869_-5.gif)
is etale. We say that
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is
totally ramified if
![$f=1$](./latex/latex2png-ClassFieldTheory_33780756_-4.gif)
. Notice that
![$e$](./latex/latex2png-ClassFieldTheory_41907218_0.gif)
and
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
are multiplicative in towers
![$K\subseteq L\subseteq L'$](./latex/latex2png-ClassFieldTheory_197029274_-3.gif)
. So
![$L'/K$](./latex/latex2png-ClassFieldTheory_171799610_-5.gif)
is unramified (resp., totally ramified) if and only if
![$L'/L$](./latex/latex2png-ClassFieldTheory_171865146_-5.gif)
and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
are unramified (resp. totally ramified).
We say
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is
tamely ramified if
![$\Char(k)\nmid e(L/K)$](./latex/latex2png-ClassFieldTheory_95298712_-5.gif)
and
![$l/k$](./latex/latex2png-ClassFieldTheory_120943636_-5.gif)
is separable, and is
wildly ramified otherwise. In particular, every unramified extension is tamely ramified.
Local fields
A
local field is a locally compact complete valued field
![$(K,|\cdot|)$](./latex/latex2png-ClassFieldTheory_239932421_-5.gif)
. So
![$(K,+)$](./latex/latex2png-ClassFieldTheory_19171714_-5.gif)
and
![$(K,\times)$](./latex/latex2png-ClassFieldTheory_123103977_-5.gif)
are locally compact Hausdorff topological group.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a non-archimedean local field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
be a finite Galois extension with
![$G=\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_15982207_-5.gif)
.
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
acts on
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
by isometries, so
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
preserves
![$\mathfrak{m}_L^i$](./latex/latex2png-ClassFieldTheory_86705309_-5.gif)
for any
![$i$](./latex/latex2png-ClassFieldTheory_42169362_0.gif)
and the action induces a map
![$G\rightarrow\Aut(\mathcal{O}_L/\mathfrak{m}_L^{i})$](./latex/latex2png-ClassFieldTheory_222515231_-5.gif)
. We define
![$G_i$](./latex/latex2png-ClassFieldTheory_221475860_-2.gif)
to be the kernel of the map
![$G\rightarrow\Aut(\mathcal{O}_L/\mathfrak{m}_L^{i+1})$](./latex/latex2png-ClassFieldTheory_129450978_-5.gif)
, called
higher ramification groups (in the lower numbering). The higher ramification groups give a filtration
![$G_0=\ker G\rightarrow \Aut(l/k)$](./latex/latex2png-ClassFieldTheory_174400994_-5.gif)
is called the
inertia group. We have
![$G/G_0=\Gal(L^\mathrm{ur}/K)$](./latex/latex2png-ClassFieldTheory_69966739_-5.gif)
,
![$G_0=\Gal(L/L^\mathrm{ur})$](./latex/latex2png-ClassFieldTheory_128591090_-5.gif)
. In particular,
![$\#G_0=e(L/K)$](./latex/latex2png-ClassFieldTheory_72501960_-5.gif)
and
![$L/L^\mathrm{ur}$](./latex/latex2png-ClassFieldTheory_42469438_-5.gif)
is totally ramified.
The fixed field of
![$G_1$](./latex/latex2png-ClassFieldTheory_220427284_-2.gif)
is the maximal tamely ramified sub-extension
![$L^\mathrm{t}$](./latex/latex2png-ClassFieldTheory_203445385_0.gif)
and
![$e(L/L^\mathrm{t})$](./latex/latex2png-ClassFieldTheory_76105790_-5.gif)
is the prime-to-
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
part of
![$e(L/K)$](./latex/latex2png-ClassFieldTheory_203711474_-5.gif)
.
![$G_0/G_1\cong \Gal(L^\mathrm{t}/L^\mathrm{ur})$](./latex/latex2png-ClassFieldTheory_40962374_-5.gif)
is called the
tame inertia group and
![$G_1\cong\Gal(L/L^\mathrm{t})$](./latex/latex2png-ClassFieldTheory_11516916_-5.gif)
is called the
wild inertia group.
![$G_0/G_1$](./latex/latex2png-ClassFieldTheory_47840971_-5.gif)
is cyclic of order prime to
![$p=\Char(k)$](./latex/latex2png-ClassFieldTheory_182551729_-5.gif)
and
![$G_1$](./latex/latex2png-ClassFieldTheory_220427284_-2.gif)
is the unique
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
-Sylow subgroup of
![$G_0$](./latex/latex2png-ClassFieldTheory_220361748_-3.gif)
.
For
![$u\ge-1$](./latex/latex2png-ClassFieldTheory_268198590_-3.gif)
we define a function
![$\phi(u)=\int_0^u\frac{dt}{[G_0:G_t]}$](./latex/latex2png-ClassFieldTheory_551616_-9.gif)
, where
![$G_u=G_{\lceil u\rceil}$](./latex/latex2png-ClassFieldTheory_53387708_-6.gif)
,
![$[G_0:G_{-1}]:=[G_{-1}:G_0]^{-1}$](./latex/latex2png-ClassFieldTheory_161779997_-5.gif)
and
![$[G_0:G_t]:=1$](./latex/latex2png-ClassFieldTheory_187130238_-5.gif)
for
![$t\in (-1,0]$](./latex/latex2png-ClassFieldTheory_245041468_-5.gif)
. Then
![$\phi(u)$](./latex/latex2png-ClassFieldTheory_50457386_-5.gif)
is a piecewise linear continuous increasing convex-up function. We define
![$\psi(v)$](./latex/latex2png-ClassFieldTheory_49408799_-5.gif)
to be the inverse function of
![$\phi(u)$](./latex/latex2png-ClassFieldTheory_50457386_-5.gif)
and define the
higher ramification groups (in the upper numbering)
![$G^v=G_{\phi(v)}$](./latex/latex2png-ClassFieldTheory_238226823_-6.gif)
. Then
![$G^{-1}=G_{-1}=G$](./latex/latex2png-ClassFieldTheory_263050664_-2.gif)
and
![$G^0=G_0$](./latex/latex2png-ClassFieldTheory_47906522_-3.gif)
, and
![$G^v=1$](./latex/latex2png-ClassFieldTheory_234654049_-1.gif)
for
![$v\gg 0$](./latex/latex2png-ClassFieldTheory_248341166_-2.gif)
. For more details, see Chapter 4 of Serre's
local fields.
Topological groups
A continuous map
![$f:X\rightarrow Y$](./latex/latex2png-ClassFieldTheory_112759265_-4.gif)
of topological space is called
proper if
![$f^{-1}(C)$](./latex/latex2png-ClassFieldTheory_18896360_-5.gif)
is compact for any
![$C $](./latex/latex2png-ClassFieldTheory_139621394_-1.gif)
compact,
open if
![$f(U)$](./latex/latex2png-ClassFieldTheory_226194490_-5.gif)
is open for any
![$U$](./latex/latex2png-ClassFieldTheory_43480082_-1.gif)
open,
closed if
![$f(C)$](./latex/latex2png-ClassFieldTheory_207320122_-5.gif)
is closed for
![$C $](./latex/latex2png-ClassFieldTheory_139621394_-1.gif)
any closed.
- A continuous map between locally compact Hausdorff spaces is proper if and only if it is closed with compact fiber.
- If
and
are proper and
is Hausdorff, then
is proper.
- If
and
are open, then
is open.
09/14/2012
A topological group is a group with a topology under which the multiplication and the addition are continuous.
The following basic properties of topological groups are easy exercises.
For any subgroup
![$H<G$](./latex/latex2png-ClassFieldTheory_201946132_-1.gif)
, we endow the coset space
![$G/H$](./latex/latex2png-ClassFieldTheory_171602964_-5.gif)
the
quotient topology, i.e.,
![$\bar U\subseteq G/H$](./latex/latex2png-ClassFieldTheory_185031136_-5.gif)
is open if and only if
![$\pi^{-1}(\bar U)$](./latex/latex2png-ClassFieldTheory_101539939_-5.gif)
is open. By definition, if
![$f: G\rightarrow G'$](./latex/latex2png-ClassFieldTheory_268066804_-4.gif)
is continuous homomorphism, then
![$G/\ker(f)\rightarrow G'$](./latex/latex2png-ClassFieldTheory_105386916_-5.gif)
is continuous.
- Let
be a surjective continuous homomorphism. Then
is a quotient map if and only if
is open.
- Let
be a closed continuous homomorphism of Hausdorff topological groups, then
is open if and only if
is open.
- Let
be a surjective continuous homomorphism and
. Suppose
is a continuous homomorphism such that
. Then
is a homeomorphism and
is a quotient map. In particular, when
is abelian, we have an isomorphism of topological groups
.
Let
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
be a locally compact Hausdorff topological group. A
Haar measure on
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is a nonzero Radon measure
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
on
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
such that
![$\mu(aU)=\mu(U)$](./latex/latex2png-ClassFieldTheory_110923483_-5.gif)
for any
![$a\in G$](./latex/latex2png-ClassFieldTheory_129393662_-1.gif)
and any measurable subset
![$U$](./latex/latex2png-ClassFieldTheory_43480082_-1.gif)
of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
.
(Haar)
The Haar measure exists and is unique up to scalar multiplication.
The Lebesgue measure on
![$\mathbb{R}$](./latex/latex2png-ClassFieldTheory_41964624_0.gif)
satisfies
![$\mu([a,b])=b-a$](./latex/latex2png-ClassFieldTheory_184000025_-5.gif)
and is a Haar measure. The standard measure on
![$\mathbb{C}\cong\mathbb{R}^2$](./latex/latex2png-ClassFieldTheory_47009391_-1.gif)
is a Haar measure.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be any local field and
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
be a Haar measure on
![$(K,+)$](./latex/latex2png-ClassFieldTheory_19171714_-5.gif)
. For
![$a\in K^\times$](./latex/latex2png-ClassFieldTheory_22902111_-1.gif)
, set
![$\mu_a(U)=\mu(aU)$](./latex/latex2png-ClassFieldTheory_64835868_-5.gif)
, then
![$\mu_a$](./latex/latex2png-ClassFieldTheory_53180082_-4.gif)
is also a Haar measure. It follows from Haar's theorem that
![$\mu_a=c\cdot \mu$](./latex/latex2png-ClassFieldTheory_237408509_-4.gif)
, where
![$c$](./latex/latex2png-ClassFieldTheory_41776146_0.gif)
is a constant.
![$\mu(aU)=||a||_K\cdot \mu(U)$](./latex/latex2png-ClassFieldTheory_111522524_-5.gif)
.
The archimedean case is obvious. Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is non-archimedean. Since
![$\mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_113154998_-2.gif)
is compact open, we know that
![$\mu(\mathcal{O}_K)$](./latex/latex2png-ClassFieldTheory_31175777_-5.gif)
is a positive real number. Thus it is enough to show that
![$\mu(a\cdot \mathcal{O}_K)=||a||_K\cdot\mu(\mathcal{O}_K)$](./latex/latex2png-ClassFieldTheory_153412637_-5.gif)
, Replacing
![$a$](./latex/latex2png-ClassFieldTheory_41645074_0.gif)
by
![$a^{-1}$](./latex/latex2png-ClassFieldTheory_6790149_0.gif)
, we may assume that
![$a\in \mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_20036107_-2.gif)
. It easy to see (via the filtration
![$\{\mathfrak{m}_K^n\}$](./latex/latex2png-ClassFieldTheory_44564416_-5.gif)
) that
![$\mathcal{O}_K/(a)=(\#k)^{\ord(a)}=||a||_K^{-1}$](./latex/latex2png-ClassFieldTheory_148081062_-5.gif)
. Hence
![$\mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_113154998_-2.gif)
is a disjoint union of
![$||a||_K^{-1}$](./latex/latex2png-ClassFieldTheory_128903090_-5.gif)
cosets of
![$\mathcal{O}_K/(a)$](./latex/latex2png-ClassFieldTheory_200559956_-5.gif)
. Therefore
![$\mu(\mathcal{O}_K)=||a||_K^{-1}\mu(a \mathcal{O}_K)$](./latex/latex2png-ClassFieldTheory_76117200_-5.gif)
using the left-invariance of
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
.
¡õ
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be local field and
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
be a Haar measure. Then
![$d\mu/||\cdot||_K$](./latex/latex2png-ClassFieldTheory_67470707_-5.gif)
is a Haar measure on
![$K^\times$](./latex/latex2png-ClassFieldTheory_112818142_0.gif)
, i.e.,
![$f\mapsto \int_{K^\times}f(x)\frac{d\mu(x)}{||x||_K}$](./latex/latex2png-ClassFieldTheory_34861058_-9.gif)
is left-invariant.
Profinite groups
A
profinite group is a (filtered) inverse limit of finite groups
![$G=\varprojlim G_i$](./latex/latex2png-ClassFieldTheory_55964281_-10.gif)
. We endow
![$\prod G_i$](./latex/latex2png-ClassFieldTheory_265568960_-4.gif)
with the product topology, which makes it a compact and Hausdorff topological group. Then
![$G\subseteq \prod G_i$](./latex/latex2png-ClassFieldTheory_107662839_-4.gif)
is a closed subspace, hence is also compact and Hausdorff and becomes a topological group under the subspace topology (equivalently, the weakest topology such that the projections
![$\pi_i: G\rightarrow G_i$](./latex/latex2png-ClassFieldTheory_151464833_-2.gif)
are continuous). Moreover, if
![$g\ne g'$](./latex/latex2png-ClassFieldTheory_199493349_-4.gif)
then there exists a projection
![$\pi_i(g)\ne \pi_i(g')$](./latex/latex2png-ClassFieldTheory_137363415_-5.gif)
, hence
![$\pi_i^{-1}(\pi_i(g))$](./latex/latex2png-ClassFieldTheory_166744368_-5.gif)
and
![$\pi_i^{-1}(\pi_i(g))$](./latex/latex2png-ClassFieldTheory_166744368_-5.gif)
are disjoint open and closed subsets, we conclude that
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is
totally disconnected.
In fact, we can characterize the profinite groups topologically as follows.
A topological group
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is profinite if and only if it is compact and totally disconnected. In this case,
![$G\cong \varprojlim G/U$](./latex/latex2png-ClassFieldTheory_220961193_-10.gif)
, where
![$U$](./latex/latex2png-ClassFieldTheory_43480082_-1.gif)
runs over all open normal subgroups of
Let
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
be any group. Then the
profinite completion of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is defined to be
![$\hat G:=\varprojlim G/H$](./latex/latex2png-ClassFieldTheory_69816891_-10.gif)
, where
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
runs over all normal subgroup of finite index of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
.
![$\hat G$](./latex/latex2png-ClassFieldTheory_28729835_-1.gif)
is profinite and the natural map
![$G\rightarrow \hat G$](./latex/latex2png-ClassFieldTheory_172219752_-1.gif)
has dense image. Any homomorphism of
![$G\rightarrow G'$](./latex/latex2png-ClassFieldTheory_132898828_-1.gif)
, for
![$G'$](./latex/latex2png-ClassFieldTheory_144274450_-1.gif)
profinite, factors through
![$G\rightarrow \hat G$](./latex/latex2png-ClassFieldTheory_172219752_-1.gif)
.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a field, then
![$G_K=\Gal(\bar K/K)$](./latex/latex2png-ClassFieldTheory_59720929_-5.gif)
is a profinite group. Hence any homomorphism
![$\mathbb{Z}\rightarrow G_K$](./latex/latex2png-ClassFieldTheory_74162620_-2.gif)
factors as
![$\mathbb{Z}\rightarrow \hat{\mathbb{Z}}\rightarrow G_K$](./latex/latex2png-ClassFieldTheory_32074594_-2.gif)
.
![$\hat{\mathbb{Z}}\cong\prod_p \mathbb{Z}_p$](./latex/latex2png-ClassFieldTheory_147436184_-8.gif)
.
For any non-archimedean local field
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
,
![$\mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_113154998_-2.gif)
and
![$\mathcal{O}_K ^\times$](./latex/latex2png-ClassFieldTheory_228669349_-5.gif)
are profinite.
If
![$\{G_i\}_{i\in I}$](./latex/latex2png-ClassFieldTheory_154795286_-5.gif)
are profinite, then the product
![$\prod_{i\in I}G_i$](./latex/latex2png-ClassFieldTheory_121936658_-7.gif)
is also profinite.
09/17/2012
Infinite Galois theory
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a field. We say
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is
separably closed if there is no finite separable extension of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We define the
separable closure of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
to be an algebraic field extension of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
which is separably closed. Any two separable closure are isomorphic.
The
absolute Galois group of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is defined to be
![$G_K=\Gal(K^\mathrm{sep}/K)$](./latex/latex2png-ClassFieldTheory_141240008_-5.gif)
.
(infinite Galois theory)
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is Galois. Then there is a bijection between subextension
![$L'$](./latex/latex2png-ClassFieldTheory_149517330_0.gif)
and
closed subgroups of
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
under the Krull topology given by
![$L'\mapsto \Gal(L/L')$](./latex/latex2png-ClassFieldTheory_28256266_-5.gif)
and
![$L^H\mapsfrom H$](./latex/latex2png-ClassFieldTheory_136811421_-1.gif)
. Moreover,
![$L'/K$](./latex/latex2png-ClassFieldTheory_171799610_-5.gif)
is Galois if and only if
![$\Gal(L/L')\lhd \Gal(L/K)$](./latex/latex2png-ClassFieldTheory_27577182_-5.gif)
is normal. In this case, we have
![$\Gal(L/K)/\Gal(L/L')\cong\Gal(L'/K)$](./latex/latex2png-ClassFieldTheory_103918273_-5.gif)
as topological groups.
We say
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is
abelian if it is Galois with abelian Galois group
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
. In this case, it is easy to see that any subextension of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian.
Let
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
be a topological group. Then
![$\overline{[G,G]}$](./latex/latex2png-ClassFieldTheory_173299411_-5.gif)
is a normal topological subgroup. The
topological abelianization of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is defined to be the Hausdorff topological group
![$G^\mathrm{ab}:=G/\overline{[G,G]}$](./latex/latex2png-ClassFieldTheory_23177548_-5.gif)
, i.e., the maximal Hausdorff abelian quotient of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
.
Now let
be a global field and
be a (possibly infinite) Galois extension.
For any
![$v\in V_K$](./latex/latex2png-ClassFieldTheory_60378006_-2.gif)
, there exists a place
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
of
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
such that
![$w|v$](./latex/latex2png-ClassFieldTheory_191859692_-5.gif)
and
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
acts transitively on such
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
.
The finite case is known. Assume that
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is infinite. We have a
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
-algebra homomorphism
![$L\rightarrow (K_v)^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_14382248_-5.gif)
induced by
![$K^\mathrm{sep}\rightarrow (K_v)^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_53240288_-5.gif)
. There is a unique absolute value
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
on
![$(K_v)^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_25756590_-5.gif)
extending
![$||\cdot||_v$](./latex/latex2png-ClassFieldTheory_203397930_-5.gif)
and restricting
![$|\cdot|$](./latex/latex2png-ClassFieldTheory_202968593_-5.gif)
to
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
gives a
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
. Now suppose
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
and
![$w'$](./latex/latex2png-ClassFieldTheory_152663058_0.gif)
are two such places of
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
. Then
![$\{g\in \Gal(L'/K)\colon g(w|_{L'})=w'|_{L'}\}$](./latex/latex2png-ClassFieldTheory_43680731_-5.gif)
is nonempty. So the inverse limit with respect to finite Galois extensions
![$L'$](./latex/latex2png-ClassFieldTheory_149517330_0.gif)
is again nonempty.
¡õ
The
decomposition group at
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is defined to be the stabilizer
![$D(w/v)=\{g\in \Gal(L/K): gw=w\}=\varprojlim D(w|_{L'}/v)$](./latex/latex2png-ClassFieldTheory_47206524_-10.gif)
. This is a closed subgroup of
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
.
Suppose
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is non-archimedean. Let
![$k_w$](./latex/latex2png-ClassFieldTheory_155284500_-2.gif)
be the the residue field of
![$(L,|\cdot|)$](./latex/latex2png-ClassFieldTheory_240980997_-5.gif)
representing
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
. This is an algebraic extension of
![$k_v$](./latex/latex2png-ClassFieldTheory_155218964_-2.gif)
. Then
![$D(w/v)$](./latex/latex2png-ClassFieldTheory_67869823_-5.gif)
acts on
![$k_w/k_v$](./latex/latex2png-ClassFieldTheory_113050284_-5.gif)
and give a continuous homomorphism
![$D(w/v)\rightarrow \Gal(k_w/k_v)$](./latex/latex2png-ClassFieldTheory_5911965_-5.gif)
. This map at each finite level is surjective, hence the image is dense. But
![$D(w/v)$](./latex/latex2png-ClassFieldTheory_67869823_-5.gif)
is compact, so this map is actually surjective. The kernel
![$I(w/v)$](./latex/latex2png-ClassFieldTheory_67869903_-5.gif)
of this map is called the
inertia group. It is a closed subgroup of
![$D(w/v)$](./latex/latex2png-ClassFieldTheory_67869823_-5.gif)
and is equal to
![$\varprojlim I(w|_{L'}/v)$](./latex/latex2png-ClassFieldTheory_264875182_-10.gif)
. We say
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is
unramified in
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
if
![$I(w/v)=\{1\}$](./latex/latex2png-ClassFieldTheory_71008620_-5.gif)
for any
![$w|v$](./latex/latex2png-ClassFieldTheory_191859692_-5.gif)
. In this case, by surjectivity, we have a
Frobenius element ![$\Frob_{w/v}\in D(w/v)$](./latex/latex2png-ClassFieldTheory_230769476_-6.gif)
whose image is
![$x\mapsto x^{\# k_v}\in \Gal(k_w/k_v)$](./latex/latex2png-ClassFieldTheory_68217326_-5.gif)
. When
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian, this doesn't depend on
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
and we have a Frobenius element
![$\Frob_v\in D(v)$](./latex/latex2png-ClassFieldTheory_44950971_-5.gif)
.
Define
![$L_w^\mathrm{alg}=\{x\in L_w: x \text{ is separable and algebraic over } K_v\}$](./latex/latex2png-ClassFieldTheory_211958361_-5.gif)
. Then
![$L_w^\mathrm{alg}=\cup K_{v'}'$](./latex/latex2png-ClassFieldTheory_62793084_-5.gif)
, where
![$K'$](./latex/latex2png-ClassFieldTheory_148468754_0.gif)
runs over all finite separable extensions of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. Moreover,
![$L_w^\mathrm{alg}/K_v$](./latex/latex2png-ClassFieldTheory_197438109_-5.gif)
is Galois and the composition
![$$D(w/v)\hookrightarrow \Aut_\mathrm{cont}(L_w/K_v)\twoheadrightarrow \Gal(L_w^\mathrm{alg}/K_v)$$](./latex/latex2png-ClassFieldTheory_53094681_.gif)
is an isomorphism of topological groups, with all maps being bijective.
Let
![$x\in L_w^\mathrm{alg}$](./latex/latex2png-ClassFieldTheory_111995298_-4.gif)
, there exists
![$x_1,x_2,\ldots \in L$](./latex/latex2png-ClassFieldTheory_43770861_-4.gif)
such that
![$x_i\rightarrow x$](./latex/latex2png-ClassFieldTheory_112252273_-2.gif)
. By Krasner's lemma, for
![$i\gg0$](./latex/latex2png-ClassFieldTheory_52590233_-2.gif)
,
![$K_v(x)\subseteq K_v(x_i)$](./latex/latex2png-ClassFieldTheory_198595810_-5.gif)
. Set
![$K'{}=K(x_i)$](./latex/latex2png-ClassFieldTheory_251394575_-5.gif)
. Then
![$K_v'\supseteq K_v(x_i)\supseteq K_v(x)$](./latex/latex2png-ClassFieldTheory_248409258_-5.gif)
. Now since
![$L=\cup K'$](./latex/latex2png-ClassFieldTheory_124363795_-1.gif)
, where
![$K'$](./latex/latex2png-ClassFieldTheory_148468754_0.gif)
runs over finite Galois subextensions, we know that any
![$K'_{v'}/K_v$](./latex/latex2png-ClassFieldTheory_253031644_-5.gif)
is Galois, hence
![$L_w^\mathrm{alg}/K_v$](./latex/latex2png-ClassFieldTheory_197438109_-5.gif)
is Galois. The composition map
![$D(w/v)\rightarrow \Gal(L_w^\mathrm{alg}/K_v)$](./latex/latex2png-ClassFieldTheory_96973936_-5.gif)
is an inverse limit of isomorphisms, hence is a topological isomorphism. The injectivity of
![$\Aut_\mathrm{cont}(L_w/K_v)$](./latex/latex2png-ClassFieldTheory_104424509_-5.gif)
follows from the fact that
![$L_w^\mathrm{alg}\supseteq L$](./latex/latex2png-ClassFieldTheory_58716487_-4.gif)
is dense in
![$L_w$](./latex/latex2png-ClassFieldTheory_230591468_-2.gif)
.
¡õ
09/19/2012
Suppose
![$L=K^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_85932851_0.gif)
. Then
![$L_w^\mathrm{alg}=K_v^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_266405548_-4.gif)
.
By definition,
![$L_w^\mathrm{alg}\subseteq K_v^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_252714799_-4.gif)
. Let
![$f\in K_v[X]$](./latex/latex2png-ClassFieldTheory_82114856_-5.gif)
be a monic irreducible separable polynomial and
![$\alpha$](./latex/latex2png-ClassFieldTheory_205233679_0.gif)
be a root of
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
. Let
![$g\in K[X]$](./latex/latex2png-ClassFieldTheory_211519081_-5.gif)
with coefficients close to
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
and the same degree as
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
. Then
![$|g(\alpha)|=|g(\alpha)-f(\alpha)|$](./latex/latex2png-ClassFieldTheory_163972095_-5.gif)
is small. On the other hand, write
![$g(X)=\prod (X-\beta_j)$](./latex/latex2png-ClassFieldTheory_237988093_-5.gif)
, where
![$\beta_j\in \overline{K_v}$](./latex/latex2png-ClassFieldTheory_255510971_-5.gif)
. Then
![$|g(\alpha)|=\prod|\alpha-\beta_j|$](./latex/latex2png-ClassFieldTheory_188803015_-5.gif)
can be made as small as possible. Now Krasner's lemma implies that
![$K_v(\alpha)\subseteq K_v(\beta_j)$](./latex/latex2png-ClassFieldTheory_241142826_-5.gif)
. Comparison of degrees shows that
![$K_v(\alpha)=K_v(\beta_j)$](./latex/latex2png-ClassFieldTheory_84853266_-5.gif)
. So
![$g$](./latex/latex2png-ClassFieldTheory_42038290_-4.gif)
is separable and irreducible over
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
, hence over
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We conclude that
![$K(\beta_j)\subseteq K^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_149325309_-5.gif)
, thus
![$\alpha\in K_v(\beta_j)\subseteq L_w^\mathrm{alg}$](./latex/latex2png-ClassFieldTheory_15034222_-5.gif)
.
¡õ
![$D(w/v)\cong G_{K_v}$](./latex/latex2png-ClassFieldTheory_156564916_-5.gif)
.
![$N_{L/K}$](./latex/latex2png-ClassFieldTheory_200669095_-6.gif)
is always open.
![$N_{L/K}$](./latex/latex2png-ClassFieldTheory_200669095_-6.gif)
is proper, hence is closed. It is enough it show that
![$N_{L/K}(L^\times)$](./latex/latex2png-ClassFieldTheory_62784245_-6.gif)
is open is
![$K^\times$](./latex/latex2png-ClassFieldTheory_112818142_0.gif)
by Exercise
3. Since we already know that the image is closed, it suffices to show that
![$N_{L/K}(L^\times)$](./latex/latex2png-ClassFieldTheory_62784245_-6.gif)
is of finite index. This follows from the norm-index formula. In fact, homological methods can show that
![$[K^\times: N_{L/K}(L^\times)]\mid [L:K]$](./latex/latex2png-ClassFieldTheory_184818406_-6.gif)
(and equality holds if and only
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian).
¡õ
The unramified case is relatively simpler.
If
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is unramified, then
![$N_{L/K}(\mathcal{O}_L^\times)=\mathcal{O}_K^\times$](./latex/latex2png-ClassFieldTheory_136513422_-6.gif)
. Hence
![$[K^\times: N_{L/K}(L^\times)]=[L:K]$](./latex/latex2png-ClassFieldTheory_142352851_-6.gif)
.
![$N(\mathcal{O}_L^\times)\subseteq \mathcal{O}_K^\times$](./latex/latex2png-ClassFieldTheory_200347519_-5.gif)
is clear. Since the image is closed in
![$\mathcal{O}_K^\times$](./latex/latex2png-ClassFieldTheory_112314149_-5.gif)
, it suffices to show that the image is dense. Any
![$g\in \Gal(L/K)$](./latex/latex2png-ClassFieldTheory_205277647_-5.gif)
acts by isometry on
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
,
![$g(1+\mathfrak{m}_L^r)\subseteq (1+\mathfrak{m}_L^r)$](./latex/latex2png-ClassFieldTheory_229324440_-5.gif)
. So
![$N(1+x)=\prod_g g(1+x)\in (1+\mathfrak{m}_L^r)\cap K^\times=1+\mathfrak{m}_K^r$](./latex/latex2png-ClassFieldTheory_161022205_-8.gif)
, where the last equality holds because
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is unramified. The norm map induces a map
![$N: \mathcal{O}_L^\times/(1+\mathfrak{m}_L)\cong l^\times\rightarrow \mathcal{O}_K^\times/(1+\mathfrak{m}_K)\cong k^\times$](./latex/latex2png-ClassFieldTheory_118710858_-5.gif)
, which is coincides with the norm map on the residue fields. By a counting argument we know that the norm map on finite fields is surjective. The norm map also induces a map
![$N: (1+\mathfrak{m}_L^r)/(1+\mathfrak{m}_L^{r+1})\cong l\rightarrow (1+\mathfrak{m}_K^r)/(1+\mathfrak{m}_K^{r+1})\cong k$](./latex/latex2png-ClassFieldTheory_163721551_-5.gif)
, which coincides with the trace map, so it is nonzero and
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
-linear (trace is always surjective for finite separable extension). This concludes that the norm map has dense image.
¡õ
Adeles
Let
![$(X_v){v\in V}$](./latex/latex2png-ClassFieldTheory_163969669_-5.gif)
be a family of topological spaces,
![$U_v\subseteq X_v$](./latex/latex2png-ClassFieldTheory_10925970_-3.gif)
be open subset defined for almost all
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
. We define
![$$X=\{(x_v)_{v\in V}: x_v\in U_v \text{ for a.a. }v \}\subseteq \prod X_v,$$](./latex/latex2png-ClassFieldTheory_132280042_.gif)
and endow
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
with the topology given by the base of open subsets
![$$\Big\{\prod_v Y_v: Y_v\subseteq X_v \text{ open}; Y_v= U_v \text{ for a.a. } v\Big\}.$$](./latex/latex2png-ClassFieldTheory_209254312_.gif)
The topological space
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
is called the
restricted topological product of
![$(X_v)$](./latex/latex2png-ClassFieldTheory_67860095_-5.gif)
with respect to
![$(U_v)$](./latex/latex2png-ClassFieldTheory_200575361_-5.gif)
. Notice that this topology is different from the subspace topology induced from the product topology.
The following lemma is easy to check.
Let
![$S\subseteq V$](./latex/latex2png-ClassFieldTheory_207184731_-3.gif)
be a finite set of indices containing all
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
's such that
![$U_v$](./latex/latex2png-ClassFieldTheory_79662060_-2.gif)
is not defined. Then
![$$X_S=\prod_{v\in S}X_v\times \prod_{v\not\in S}U_v$$](./latex/latex2png-ClassFieldTheory_22425354_.gif)
is open in
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
and the subspace topology on
![$X_S\subseteq X$](./latex/latex2png-ClassFieldTheory_207326486_-3.gif)
is the product topology on
![$X_S$](./latex/latex2png-ClassFieldTheory_29002732_-3.gif)
.
The following proposition partially explains the reason of introducing the restricted product.
If
![$X_v$](./latex/latex2png-ClassFieldTheory_29330412_-2.gif)
's are locally compact Hausdorff and the
![$U_v$](./latex/latex2png-ClassFieldTheory_79662060_-2.gif)
's are compact, then
![$X$](./latex/latex2png-ClassFieldTheory_43676690_0.gif)
is locally compact Hausdorff.
Notice that
![$X_S$](./latex/latex2png-ClassFieldTheory_29002732_-3.gif)
is locally compact: it is a product of a locally compact Hausdorff space and a compact space. Then result then follows from the fact that
![$X=\bigcup_S X_S$](./latex/latex2png-ClassFieldTheory_58207879_-6.gif)
.
¡õ
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field. The
ring of ideles ![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
is defined to be the restricted product of
![$\{K_v\}_{v\in V_K}$](./latex/latex2png-ClassFieldTheory_44273113_-5.gif)
with respect to
![$\{\mathcal{O}_v\}$](./latex/latex2png-ClassFieldTheory_178691307_-5.gif)
for
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
non-archimedean. It is a subring of
![$\prod_v K_v$](./latex/latex2png-ClassFieldTheory_233888172_-5.gif)
and is a locally compact Hausdorff topological ring.
Since
![$\hat{\mathbb{Z}}\cong\prod_p \mathbb{Z}_p$](./latex/latex2png-ClassFieldTheory_147436184_-8.gif)
, we have
![$\mathbb{A}_\mathbb{Q}=\mathbb{R}\times (\hat{\mathbb{Z}}\otimes _\mathbb{Z}\mathbb{Q})$](./latex/latex2png-ClassFieldTheory_97239511_-5.gif)
. The similar identification works for general number fields when replacing
![$\hat{\mathbb{Z}}$](./latex/latex2png-ClassFieldTheory_6257619_0.gif)
by
![$\widehat{\mathcal{O}_K}$](./latex/latex2png-ClassFieldTheory_175700012_-2.gif)
. We also have
![$\mathbb{A}_K=\mathbb{A}_\mathbb{Q}\otimes K$](./latex/latex2png-ClassFieldTheory_202122912_-4.gif)
.
![$K\subseteq \mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_51528911_-3.gif)
is a discrete closed subgroup and the quotient
![$\mathbb{A}_K/K$](./latex/latex2png-ClassFieldTheory_25455226_-5.gif)
is compact.
09/21/2012
We will omit the tedious measure-theoretic check of the following the lemma.
There exists a Haar measure
![$\mu_K$](./latex/latex2png-ClassFieldTheory_54359730_-4.gif)
on
![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
such that
![$\mu_K(\prod U_v)=\prod \mu_v(U_v)$](./latex/latex2png-ClassFieldTheory_27785565_-5.gif)
where
![$\mu_v$](./latex/latex2png-ClassFieldTheory_54556338_-4.gif)
is the normalized Haar measure on
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
such that
![$\mu_v(\mathcal{O}_v)=1$](./latex/latex2png-ClassFieldTheory_1365072_-5.gif)
.
Ideles
The group of
ideles is defined to be
![$\mathbb{A}_K^\times=\{ x\in \mathbb{A}_K: x_v\ne0; ||x_v||_v=1\text{ for a.a. } v\}$](./latex/latex2png-ClassFieldTheory_124541186_-5.gif)
. The embeddings
![$K_v^\times\hookrightarrow \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_102090037_-5.gif)
defined by
![$x\mapsto (1,\ldots1, x,1,\ldots,1)$](./latex/latex2png-ClassFieldTheory_235535618_-5.gif)
gives an embedding
![$K^\times\hookrightarrow \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_116013117_-5.gif)
. The quotient
![$\mathbb{A}_K^\times/ K^\times$](./latex/latex2png-ClassFieldTheory_112559470_-5.gif)
is the called the
idele class group of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
The
idelic norm is defined to be the homomorphism
![$||\cdot||_K: \mathbb{A}_K^\times\rightarrow \mathbb{R}_{>0}$](./latex/latex2png-ClassFieldTheory_171656476_-5.gif)
given by
![$||x||_K=\prod_{v\in V_K} ||x_v||_v$](./latex/latex2png-ClassFieldTheory_38130914_-7.gif)
. The
unit ideles is defined to be the subgroup
![$(\mathbb{A}_K^\times)^1=\{x\in \mathbb{A}_K^\times: ||x||_K=1\}\subseteq \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_19068735_-5.gif)
.
is continuous. Hence
is a closed subgroup.
- When
is a number field,
is surjective and open, hence is a quotient map.
When
is a global function field, the image of
can be described as follows.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global function field with constant field
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
and
![$q=\#k$](./latex/latex2png-ClassFieldTheory_242579103_-4.gif)
. Then
![$||\mathbb{A}_K^\times||=q^\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_67534714_-5.gif)
.
For
![$x\in \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_259462054_-5.gif)
, then multiplication by
![$x$](./latex/latex2png-ClassFieldTheory_43152402_0.gif)
scales
![$\mu_K$](./latex/latex2png-ClassFieldTheory_54359730_-4.gif)
by
![$||x||_K$](./latex/latex2png-ClassFieldTheory_96465497_-5.gif)
.
It follows from the case of local fields (Lemma
1) and the Lemma
6.
¡õ
![$\frac{d\mu_K}{||\cdot||_K}$](./latex/latex2png-ClassFieldTheory_145530331_-9.gif)
is a Haar measure on
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
.
From this we can obtain a slick proof of the product formula.
09/24/2012
(Theorem 2)
Consider the measure
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
in Remark
42. Suppose
![$w\in K^\times$](./latex/latex2png-ClassFieldTheory_22902089_-1.gif)
, then
![$$||w||_K\int_{\mathbb{A}_K}f(x)d\mu(x)=\int_{\mathbb{A}_K}f(wx)d\mu(x)=\int_{\mathbb{A}_K/K}\int_K f(wyz)d\nu(y)d\bar\mu(\bar z).$$](./latex/latex2png-ClassFieldTheory_76852232_.gif)
Since
![$d\nu $](./latex/latex2png-ClassFieldTheory_183662228_0.gif)
is
invariant under multiplication by
![$w$](./latex/latex2png-ClassFieldTheory_43086866_0.gif)
, this integral is equal to
![$$\int_{\mathbb{A}_K/K}\int_K f(yz) d\nu(y)d\bar\mu(\bar z)=\int_{\mathbb{A}_K }f(x)d\mu(x).$$](./latex/latex2png-ClassFieldTheory_267597455_.gif)
Hence
![$||w||_K=1$](./latex/latex2png-ClassFieldTheory_27068588_-5.gif)
.
¡õ
Let
![$x\in (\mathbb{A}_\mathbb{Q}^\times)^1$](./latex/latex2png-ClassFieldTheory_57208459_-7.gif)
. Then the fractional ideal
![$\prod_{p}(p)^{\ord_p(x_p)}=(y)$](./latex/latex2png-ClassFieldTheory_227581338_-8.gif)
where
![$y\in \mathbb{Q}^\times$](./latex/latex2png-ClassFieldTheory_251338858_-4.gif)
determined up to sign. Hence
![$x/y\in \mathbb{R}^\times\times\hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_117691450_-5.gif)
. But
![$||x/y||_K=1$](./latex/latex2png-ClassFieldTheory_124339538_-5.gif)
implies that
![$||x_\infty/y||_{\infty}=1$](./latex/latex2png-ClassFieldTheory_173975805_-5.gif)
. Replacing
![$y$](./latex/latex2png-ClassFieldTheory_43217938_-4.gif)
by
![$-y$](./latex/latex2png-ClassFieldTheory_119763986_-4.gif)
if neccesary, there exists a unique
![$y\in \mathbb{Q}^\times$](./latex/latex2png-ClassFieldTheory_251338858_-4.gif)
such that
![$x_\infty/y=1$](./latex/latex2png-ClassFieldTheory_168367951_-5.gif)
as well. Hence
![$\{1\}\times\hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_87455153_-5.gif)
is a fundamental domain for
![$(\mathbb{A}_\mathbb{Q}^\times)^1/\mathbb{Q}^\times$](./latex/latex2png-ClassFieldTheory_191921283_-7.gif)
. Therefore
![$(\mathbb{A}_\mathbb{Q}^\times)^1/\mathbb{Q}^\times\cong \hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_42356273_-7.gif)
as topological groups. Hence
![$\mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times\cong \mathbb{R}_{>0}\times \hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_237720958_-7.gif)
as topological groups. Now
![$\mathbb{R}_{>0}\times \hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_112436695_-4.gif)
also embeds into
![$\mathbb{A}_\mathbb{Q}^\times$](./latex/latex2png-ClassFieldTheory_215282759_-7.gif)
, hence
![$\mathbb{A}_\mathbb{Q}^\times\cong \mathbb{Q}^\times\times \mathbb{R}_{>0}\times \hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_18871660_-7.gif)
as topological groups (this can also be seen directly by extracting all
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
-powers of
![$x_p$](./latex/latex2png-ClassFieldTheory_163941356_-5.gif)
).
Adelic Minkowski's theorem
The classical Minkowski's theorem says that for a compact convex and symmetric around 0 region
,
implies that there exists a nonzero
such that
. The following is a reformulation.
(Minkowski)
Let
![$V$](./latex/latex2png-ClassFieldTheory_43545618_-1.gif)
be a
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-dimensional
![$\mathbb{R}$](./latex/latex2png-ClassFieldTheory_41964624_0.gif)
-vector space and
![$\Lambda\subseteq V$](./latex/latex2png-ClassFieldTheory_160875502_-3.gif)
be a lattice. Suppose
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
is a Haar measure on
![$V$](./latex/latex2png-ClassFieldTheory_43545618_-1.gif)
constructed from the counting measure on
![$\Lambda$](./latex/latex2png-ClassFieldTheory_234755343_0.gif)
and the volume one measure on
![$V/\Lambda$](./latex/latex2png-ClassFieldTheory_258685681_-5.gif)
. If
![$S\subseteq V$](./latex/latex2png-ClassFieldTheory_207184731_-3.gif)
is compact convex and symmetric around 0, then
![$\mu(S)>2^n$](./latex/latex2png-ClassFieldTheory_36860271_-5.gif)
implies that
![$S\cap\Lambda\ne \{0\}$](./latex/latex2png-ClassFieldTheory_82191845_-5.gif)
.
The idea of the proof of the following adelic version is essentially the same as the classical version.
(Adelic Minkowski)
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field and
![$x\in \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_259462054_-5.gif)
. Then
![$$S_x:=\{y\in \mathbb{A}_K: ||y_v||_v\le ||x_v||_v\}$$](./latex/latex2png-ClassFieldTheory_250873712_.gif)
is compact. There exists
![$c>0$](./latex/latex2png-ClassFieldTheory_15567852_-1.gif)
depending only on
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
such that if
![$||x||_K>c$](./latex/latex2png-ClassFieldTheory_28772780_-5.gif)
, then
![$S_x\cap K\ne \{0\}$](./latex/latex2png-ClassFieldTheory_188976287_-5.gif)
.
(Strong approximation)
Let
![$v_0\in V_K$](./latex/latex2png-ClassFieldTheory_92263530_-3.gif)
and
![$\mathbb{A}_K^{v_0}=\mathbb{A}_K /K_{v_0}$](./latex/latex2png-ClassFieldTheory_91069918_-5.gif)
. Then the diagonal embedding
![$K\hookrightarrow \mathbb{A}_K^{v_0}$](./latex/latex2png-ClassFieldTheory_199217163_-5.gif)
has dense image.
We claim that there exists
![$w\in \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_259466150_-5.gif)
such that
![$S_w\twoheadrightarrow \mathbb{A}_K/K$](./latex/latex2png-ClassFieldTheory_20518733_-5.gif)
. This follows from the fact that
![$\mathbb{A}_K\rightarrow \mathbb{A}_K/K$](./latex/latex2png-ClassFieldTheory_21760438_-5.gif)
is open (Exercise
3),
![$\mathbb{A}_K=\bigcup_w S_w$](./latex/latex2png-ClassFieldTheory_1723967_-5.gif)
and
![$\mathbb{A}_K /K$](./latex/latex2png-ClassFieldTheory_148927661_-5.gif)
is compact.
For such a
, let
,
and
. Then for
, we choose
such that
.
and
, we let
.
, we choose
such that
has
, where
is the constant in the Adelic Minkowski's Theorem 15.
By Adelic Minkowski's Theorem 15, there exists
. Write
such that
and
. So
, then
. Then
, where
and
for any
by construction.
for
and
.
¡õ
09/26/2012
![$(\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_9756440_-5.gif)
is closed in
![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
and the subspace topology on
![$(\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_9756440_-5.gif)
from
![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
coincides with the subspace topology from
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
.
First we show that
![$(\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_9756440_-5.gif)
is closed in
![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
. For
![$x\in \mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_155994225_-2.gif)
, the products
![$\nu_N=\prod_{i=1}^N||x_{v_i}||_{v_i}$](./latex/latex2png-ClassFieldTheory_261866272_-5.gif)
will eventually be decreasing. So
![$||x||_K=\lim_{N\rightarrow\infty} \nu_N$](./latex/latex2png-ClassFieldTheory_202809393_-5.gif)
is well-defined. Suppose
![$x\in \mathbb{A}_K \setminus (\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_257167000_-5.gif)
, then
![$||x||_K\ne1$](./latex/latex2png-ClassFieldTheory_60362325_-5.gif)
. There are two cases:
. Let
be a finite set of places such that
and
for all
. For
and
, we let
and
Then
is an open neighborhood of
in
. For
small enough,
.
. Let
be a finite set of places such that
for
.
for
. This implies that if
, and
, then
.
.
For
, let
be a small open neighborhood of
small enough such that
for
. Then
is a neighborhood of
in
. Let
. If
for some
, then
. If
for all
, then
. Hence
. This concludes that
is closed in
.
Now
is continuous and has closed image. So we need to show that any neighborhood of
in
contains
for some
a neighborhood of
in
. By homogeneity, we may assume
. The basic open neighborhood of
in
is of the form
, where
and
is an open neighborhood of 1 in
. Shrinking the
's we may assume
for any
. We claim that
works, i.e.,
. If for some
, we have
, then
, thus
a contradiction.
¡õ
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field. Then
![$K^\times\hookrightarrow \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_116013117_-5.gif)
is a discrete closed subgroup and
![$(\mathbb{A}_K^\times)^1/K^\times$](./latex/latex2png-ClassFieldTheory_41433218_-5.gif)
is compact.
The assertion that
![$K^\times$](./latex/latex2png-ClassFieldTheory_112818142_0.gif)
is discrete and closed follows form the case of
![$K\hookrightarrow \mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_146284291_-2.gif)
(Theorem
12) since the topology on
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
is finer than
![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
. It remains to prove the compactness of
![$(\mathbb{A}_K^\times)^1/K^\times$](./latex/latex2png-ClassFieldTheory_41433218_-5.gif)
. By the previous lemma, if
![$W\subseteq \mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_51531983_-3.gif)
is compact, then
![$W\cap (\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_9657932_-5.gif)
is compact in
![$(\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_9756440_-5.gif)
. So it suffices to show that there is a surjection
![$W\cap (\mathbb{A}_K^\times)^1\twoheadrightarrow (\mathbb{A}_K^\times)^1/K^\times$](./latex/latex2png-ClassFieldTheory_10841189_-5.gif)
for some
![$W\subseteq \mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_51531983_-3.gif)
compact. Let
![$c>0$](./latex/latex2png-ClassFieldTheory_15567852_-1.gif)
be as in the Adelic Minkowski's Theorem
15 and choose
![$x\in \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_259462054_-5.gif)
such that
![$||x||_K>c$](./latex/latex2png-ClassFieldTheory_28772780_-5.gif)
. We claim that the compact set
![$W=S_x$](./latex/latex2png-ClassFieldTheory_155350701_-2.gif)
works. Let
![$y\in (\mathbb{A}_K^\times)^1$](./latex/latex2png-ClassFieldTheory_253059263_-5.gif)
, then
![$||x/y||_K=||x||_K>c$](./latex/latex2png-ClassFieldTheory_136687543_-5.gif)
. It follows from the Adelic Minkowski's Theorem
15 that there exists
![$z\in K^\times\cap S_{x/y}$](./latex/latex2png-ClassFieldTheory_166341632_-6.gif)
, i.e.,
![$||z||_v\le{||x/y}||_v$](./latex/latex2png-ClassFieldTheory_93065442_-5.gif)
for any
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
. Now
![$||yz||_v\le||x_v||_v$](./latex/latex2png-ClassFieldTheory_70146877_-5.gif)
, hence
![$yz\in (\mathbb{A}_K^\times)^1\cap W$](./latex/latex2png-ClassFieldTheory_153460162_-5.gif)
. This concludes the surjectivity of
![$W\cap (\mathbb{A}_K^\times)^1\twoheadrightarrow (\mathbb{A}_K^\times)^1/K^\times$](./latex/latex2png-ClassFieldTheory_10841189_-5.gif)
.
¡õ
Classical finiteness theorems
Let
![$S\supseteq S_\infty$](./latex/latex2png-ClassFieldTheory_36100596_-3.gif)
be a finite set of places. We denote
![$$\mathbb{A}_{K,S}=\prod_{v\in S} K_v\times\prod_{v\not\in S}\mathcal{O}_v,$$](./latex/latex2png-ClassFieldTheory_8927248_.gif)
an open subring of
![$\mathbb{A}_K$](./latex/latex2png-ClassFieldTheory_199380429_-2.gif)
. Similarly, we denote
![$$\mathbb{A}_{K,S}^\times==\prod_{v\in S} K_v^\times\times\prod_{v\not\in S}\mathcal{O}_v^\times,$$](./latex/latex2png-ClassFieldTheory_76045391_.gif)
an open subgroup of
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
. We have
![$\mathcal{O}_{K,S}=K\cap \mathbb{A}_{K,S}$](./latex/latex2png-ClassFieldTheory_172034014_-5.gif)
and
![$\mathcal{O}_{K,S}^\times=K^\times\cap \mathbb{A}_{K,S}^\times$](./latex/latex2png-ClassFieldTheory_139574332_-8.gif)
.
Recall that the
-class group (Definition 11) is
, the fractional ideals of
quotient by the principal ideals. We have a natural surjection
with kernel
and
. Hence we have an isomorphism ![$$\Phi: \mathbb{A}_K^\times/K^\times \mathbb{A}_{K,S}^\times\cong \Cl_S(K).$$](./latex/latex2png-ClassFieldTheory_161580207_.gif)
09/28/2012
Idele class groups
Let
be a finite extension of global fields. We know that
as topological rings. In particular,
is a closed embedding (i.e., a homeomorphism onto a closed subgroup). Hence
is also a closed embedding.
The natural map
![$\mathbb{A}_K^\times/K^\times\hookrightarrow \mathbb{A}_L^\times/L^\times$](./latex/latex2png-ClassFieldTheory_116777135_-5.gif)
is a closed embedding.
For any
![$x\in \mathbb{A}_L^\times$](./latex/latex2png-ClassFieldTheory_259457958_-5.gif)
,
![$||x||_L=||N_{L/K}(x)||_K$](./latex/latex2png-ClassFieldTheory_39181346_-6.gif)
. Hence
![$(\mathbb{A}_L^\times)^1 /L^\times\cap \mathbb{A}_K^\times/K^\times=(\mathbb{A}_K^\times)^1 /K^\times$](./latex/latex2png-ClassFieldTheory_227647394_-5.gif)
. When
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global function field, we have two exact sequences
![$$\xymatrix{0 \ar[r] & (\mathbb{A}_K^\times)^1/K^\times \ar[r]\ar@{^{(}->}[d] & \mathbb{A}_K^\times/K^\times\ar[r] \ar@{^{(}->}[d] &q^\mathbb{Z} \ar[r] \ar[d]^{[L:K]}& 0\\ 0 \ar[r] & (\mathbb{A}_L^\times)^1/L^\times \ar[r] & \mathbb{A}_L^\times/L^\times \ar[r] & q^\mathbb{Z}\ar[r]& 0.} $$](./latex/latex2png-ClassFieldTheory_200114958_.gif)
Since
![$(\mathbb{A}_K^\times)^1/K^\times$](./latex/latex2png-ClassFieldTheory_41433218_-5.gif)
is compact, the first vertical map is a closed embedding. The last vertical map is also a closed embedding. Now the middle term is infinite union of these closed embeddings, hence is also a closed embedding. The number field case is similar:
![$$\xymatrix{0 \ar[r] & (\mathbb{A}_K^\times)^1/K^\times \ar[r]\ar@{^{(}->}[d] & \mathbb{A}_K^\times/K^\times\ar[r] \ar@{^{(}->}[d] & \mathbb{R}_{>0} \ar[r] \ar[d]^{[L:K]}& 0\\ 0 \ar[r] & (\mathbb{A}_L^\times)^1/L^\times \ar[r] & \mathbb{A}_L^\times/L^\times \ar[r] & \mathbb{R}_{>0}\ar[r]& 0.} $$](./latex/latex2png-ClassFieldTheory_235775999_.gif)
The same argument shows that the middle map is a closed embedding.
¡õ
- For any
,
is a closed embedding.
- If
is a finite set of places and
. Then
is not a closed embedding.
Notice that
is a finite
-module, we obtain a norm map
, compatible with the norm
. Using
, we know that for any
, the norm map is also compatible with the local norm
. Moreover,
for any
.
![$N_{L/K}: \mathbb{A}_L^\times /L^\times\rightarrow \mathbb{A}_K^\times /K^\times$](./latex/latex2png-ClassFieldTheory_231360805_-6.gif)
is continuous, open and proper.
It is continuous since each local norm is continuous and
![$N_{L_w/K_v}^{-1}(\mathcal{O}_{K_v}^\times)=\mathcal{O}_{L_w}^\times$](./latex/latex2png-ClassFieldTheory_70011651_-9.gif)
(hence the inverse image of a basic open subset is open). For the properness, we use the splitting
![$\mathbb{A}_L^\times/L^\times$](./latex/latex2png-ClassFieldTheory_14411959_-5.gif)
and
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
. Then
![$N_{L/K}: \mathbb{A}_L^\times /L^\times\rightarrow \mathbb{A}_K^\times /K^\times$](./latex/latex2png-ClassFieldTheory_231360805_-6.gif)
is the norm on the compact factors
![$(\mathbb{A}_L^\times)^1/L^\times$](./latex/latex2png-ClassFieldTheory_226998142_-5.gif)
and identity on
![$q^\mathbb{Z} $](./latex/latex2png-ClassFieldTheory_268404096_-4.gif)
or
![$\mathbb{R}_{>0}$](./latex/latex2png-ClassFieldTheory_180773804_-4.gif)
, so it is a product of two proper maps, hence is proper. To prove the openness, we use the fact that local norms are open (Theorem
11) and the local norm is surjective for unramified local extensions (Lemma
4) to conclude that the image of a basic open subset is open.
¡õ
The map
![$\mathbb{A}_K^\times/K^\times\rightarrow \mathbb{A}_K^\times/K^\times: x\mapsto x^n$](./latex/latex2png-ClassFieldTheory_79982081_-5.gif)
is continuous and proper.
Our next goal is to describe the connected component of 1 in ideles class group
(which turns out to be exactly the kernel of the global Artin map by class field theory).
First suppose
is a number field. Then the connected component of 1 in
is
where
and
are the numbers of real and complex places of
. It is divisible, i.e.,
is surjective for any
. Let
be its closure in
. Then
is a closed connected divisible subgroup (the divisibility follows from the fact that
is proper, thus closed).
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field. Then
![$D_K$](./latex/latex2png-ClassFieldTheory_171799572_-2.gif)
is the connected component of 1 in
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
and
![$(\mathbb{A}_K^\times/K^\times)/D_K$](./latex/latex2png-ClassFieldTheory_235706809_-5.gif)
is profinite. Moreover,
![$D_K$](./latex/latex2png-ClassFieldTheory_171799572_-2.gif)
is the set of all divisible elements in
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
.
Notice that
![$\mathbb{A}_K^\times/K^\times\mathbb{A}_{K,S_\infty}^\times\cong \Cl(K)$](./latex/latex2png-ClassFieldTheory_206379569_-8.gif)
is finite and
![$[(\prod_{v|\infty} K_v^\times: (\prod_{v|\infty} K_v^\times)^0]<\infty$](./latex/latex2png-ClassFieldTheory_63543946_-8.gif)
, we know that the natural map
![$\prod_{v\nmid\infty} \mathcal{O}_v^\times\rightarrow (\mathbb{A}_K^\times/K^\times)/D_K$](./latex/latex2png-ClassFieldTheory_176521389_-8.gif)
has finite cokernel. Let
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
be the image of this last map. Since
![$\prod_{v\nmid \infty}\mathcal{O}_v^\times$](./latex/latex2png-ClassFieldTheory_236085145_-8.gif)
is profinite, we know that the image
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
is also a profinite group (Remark
30). Since
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
is of finite index, it is also open in
![$(\mathbb{A}_K^\times/K^\times)/D_K$](./latex/latex2png-ClassFieldTheory_235706809_-5.gif)
. Combining the fact that
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
is compact and totally disconnected, we find that
![$(\mathbb{A}_K^\times/K^\times)/D_K$](./latex/latex2png-ClassFieldTheory_235706809_-5.gif)
is also compact and totally disconnected, thus is profinite.
Let
be the connected component of 1, then
is killed under the map to
by totally disconnectedness, therefore
. But
is already connected, this shows that
. Every divisible element maps to 1 in
since profinite group has no nontrivial divisible elements, so must lie in
. But
is already divisible, hence it consists of all divisible elements of
.
¡õ
Now consider the global function field case.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global function field. Then
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
is totally disconnected and has no nontrivial divisible elements.
Then
![$\prod\mathcal{O}_v^\times$](./latex/latex2png-ClassFieldTheory_156419501_-4.gif)
is an open neighborhood of 1 in
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
, hence
![$\prod\mathcal{O}_v^\times$](./latex/latex2png-ClassFieldTheory_156419501_-4.gif)
is an open neighborhood of 1 in
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
. Hence
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
is totally disconnected, thus
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
is profinite. Hence
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
has no divisible elements.
¡õ
Cyclotomic extensions
Let
be any field and
. Let
be a primitive
-th root of unity. Then
, the splitting field of
, is separable, hence is Galois. Let
. Then any
is determined by its action on
, thus we obtain a injection
This in particular shows that
is abelian. The map
is functorial, i.e., for any field extension
, we have a commutative diagram ![$$\xymatrix{\Gal(L(\zeta_m)/L) \ar[r]^-{\alpha_m} \ar[d]& (\mathbb{Z}/m \mathbb{Z})^\times\\ \Gal(K(\zeta_m)/K) \ar[ru]^-{\alpha_m} &}.$$](./latex/latex2png-ClassFieldTheory_207774941_.gif)
For
![$d\mid m$](./latex/latex2png-ClassFieldTheory_239728155_-5.gif)
,
![$\zeta_d=\zeta_m^{m/d}$](./latex/latex2png-ClassFieldTheory_118874652_-4.gif)
and
![$K(\zeta_d)\subseteq K(\zeta_m)$](./latex/latex2png-ClassFieldTheory_243832832_-5.gif)
. So
![$\{K(\zeta_d)\}_{\Char(K)\nmid m}\}$](./latex/latex2png-ClassFieldTheory_46918667_-6.gif)
forms a filtered directed system. We define the
maximal cyclotomic extension to be
![$K^\mathrm{cyc}=\cup_{\Char(K)\nmid m} K(\zeta_m)$](./latex/latex2png-ClassFieldTheory_216108484_-6.gif)
.
10/01/2012
When
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
or
![$\mathbb{Q}_p$](./latex/latex2png-ClassFieldTheory_199577036_-5.gif)
, the Kronecker-Weber theorem says that
![$K^\mathrm{ab}=K^\mathrm{cyc}$](./latex/latex2png-ClassFieldTheory_146814276_0.gif)
. However, this is not true for general number fields. For example, for
![$K=\mathbb{Q}(\sqrt{2})$](./latex/latex2png-ClassFieldTheory_130230034_-5.gif)
, the extension
![$K(\sqrt[4 ]{2})$](./latex/latex2png-ClassFieldTheory_91220053_-5.gif)
is abelian over
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
but is not even Galois over
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
, hence
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
cannot be contained in a cyclotomic extension of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
since
![$K(\zeta_m)=\mathbb{Q}(\zeta_m)\cdot K$](./latex/latex2png-ClassFieldTheory_173549416_-5.gif)
is abelian over
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
for any
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
.
Suppose
![$K=k=\mathbb{F}_q$](./latex/latex2png-ClassFieldTheory_71607164_-5.gif)
, where
![$q=p^r$](./latex/latex2png-ClassFieldTheory_30640481_-4.gif)
. Suppose
![$p\nmid m$](./latex/latex2png-ClassFieldTheory_242242747_-5.gif)
and
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
is the order of
![$q\in (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_243221760_-5.gif)
. Since
![$\Frob(\zeta_m)=\zeta_m^q$](./latex/latex2png-ClassFieldTheory_110952687_-5.gif)
, we know that
![$\alpha_m(\Frob)=q\mod{m}$](./latex/latex2png-ClassFieldTheory_64701160_-5.gif)
. Therefore
![$\Gal(k(\zeta_m)/k)\cong \mathbb{Z}/f \mathbb{Z}$](./latex/latex2png-ClassFieldTheory_27381474_-5.gif)
and
![$k(\zeta_m)$](./latex/latex2png-ClassFieldTheory_69629390_-5.gif)
is a degree
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
extension of
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
. We have
![$k^\mathrm{cyc}=\bar k$](./latex/latex2png-ClassFieldTheory_220466819_0.gif)
,
![$G_k\cong\hat{ \mathbb{Z} }$](./latex/latex2png-ClassFieldTheory_235873116_-2.gif)
and
![$\alpha_k: \hat{\mathbb{Z}}\rightarrow \prod_{\ell\ne q} \mathbb{Z}_{p}^\times$](./latex/latex2png-ClassFieldTheory_255619463_-8.gif)
is given by
![$1\mapsto q$](./latex/latex2png-ClassFieldTheory_256114206_-4.gif)
.
Suppose
![$K=\mathbb{R}$](./latex/latex2png-ClassFieldTheory_192959608_0.gif)
and
![$m>2$](./latex/latex2png-ClassFieldTheory_152335380_-1.gif)
, then
![$K(\zeta_m)=\mathbb{C}$](./latex/latex2png-ClassFieldTheory_200431125_-5.gif)
and
![$\alpha_m$](./latex/latex2png-ClassFieldTheory_10391138_-2.gif)
sends the complex conjugation to
![$-1\mod{m}$](./latex/latex2png-ClassFieldTheory_99097427_0.gif)
as
![$\bar\zeta_m=\zeta_m^{-1}$](./latex/latex2png-ClassFieldTheory_215046456_-4.gif)
.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a non-archimedean local field. If
![$p\nmid m$](./latex/latex2png-ClassFieldTheory_242242747_-5.gif)
, then
![$K(\zeta_m)/K$](./latex/latex2png-ClassFieldTheory_95185874_-5.gif)
is unramified. Indeed, if
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is any finite extension, Hensel's lemma implies that
![$\zeta_m\in L$](./latex/latex2png-ClassFieldTheory_242116588_-4.gif)
if and only if
![$\zeta_m\in l$](./latex/latex2png-ClassFieldTheory_241592300_-4.gif)
. Hence
![$K(\zeta_m)$](./latex/latex2png-ClassFieldTheory_61240782_-5.gif)
is the unramified extension with residue field
![$k(\zeta_m)$](./latex/latex2png-ClassFieldTheory_69629390_-5.gif)
. We have
![$\alpha_m(\Frob)=q\mod{m}$](./latex/latex2png-ClassFieldTheory_64701160_-5.gif)
as the case of finite fields.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global function field, then
![$K^\mathrm{cyc}=K\otimes _k\bar k$](./latex/latex2png-ClassFieldTheory_139168733_-2.gif)
. However, it is not clear whether this is the maximal abelian extension (indeed, not in general).
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a number field and
![$m>1$](./latex/latex2png-ClassFieldTheory_152269844_-1.gif)
. Then
![$K(\zeta_m)$](./latex/latex2png-ClassFieldTheory_61240782_-5.gif)
is ramified at most over finite places
![$v\mid m$](./latex/latex2png-ClassFieldTheory_239732763_-5.gif)
and ramified at the real places if
![$m>2$](./latex/latex2png-ClassFieldTheory_152335380_-1.gif)
. For
![$v\nmid m$](./latex/latex2png-ClassFieldTheory_242267323_-5.gif)
,
![$\alpha_m(\Frob_v)=q_v\mod{m}$](./latex/latex2png-ClassFieldTheory_131507279_-5.gif)
, where
![$q_v=\# k_v$](./latex/latex2png-ClassFieldTheory_231434505_-4.gif)
, due to the compatibility of
![$\alpha_m$](./latex/latex2png-ClassFieldTheory_10391138_-2.gif)
with respect to the inclusion
![$K\rightarrow K_v$](./latex/latex2png-ClassFieldTheory_107443128_-2.gif)
, and the decomposition group
![$D(v)=\langle \Frob_v\rangle$](./latex/latex2png-ClassFieldTheory_260865540_-5.gif)
maps to
![$\langle q_v\rangle \subseteq (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_91366283_-5.gif)
. For
![$v|\infty$](./latex/latex2png-ClassFieldTheory_54308375_-5.gif)
,
![$D(v)=\{1,\sigma\}$](./latex/latex2png-ClassFieldTheory_132142136_-5.gif)
maps to
![$\{\pm1\}\subseteq (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_220605461_-5.gif)
for
![$m>2$](./latex/latex2png-ClassFieldTheory_152335380_-1.gif)
. The question remaining is what
![$\alpha_m$](./latex/latex2png-ClassFieldTheory_10391138_-2.gif)
takes
![$\Frob_v$](./latex/latex2png-ClassFieldTheory_264039232_-2.gif)
for
![$v|m$](./latex/latex2png-ClassFieldTheory_209226732_-5.gif)
(which can be answered by Artin reciprocity law).
Now consider the case
, we have
.
is ramified at
if and only if
(and
is even for
). In fact,
is totally ramified in
since
. Write
, then
is the compositum field of
(where
is unramified) and
(where
is totally ramified). By Chinese remainder theorem, we know that
hence these two fields are linear disjoint. Comparing ramification index shows that the prime above
of
is totally ramified in
and the prime above
in
is unramified in
. So
is the maximal unramified subextension at
in
. Hence
maps isomorphic to
under
and
given by
. Therefore
maps isomorphically to
under
.
Let
![$K\subseteq \mathbb{Q}^\mathrm{cyc}$](./latex/latex2png-ClassFieldTheory_84694294_-3.gif)
be a finite subextension. Then there exists a unique smallest
![$f=\mathfrak{f}_{K/\mathbb{Q}}$](./latex/latex2png-ClassFieldTheory_3617567_-6.gif)
(the notation comes from German word
Führer) such that
![$K\subseteq \mathbb{Q}(\zeta_f)$](./latex/latex2png-ClassFieldTheory_83437763_-5.gif)
. Moreover,
![$p|f$](./latex/latex2png-ClassFieldTheory_226522132_-5.gif)
if and only if
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
is ramified in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
Since
![$K/\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_42969992_-5.gif)
is finite, there exists an
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
such that
![$K\subseteq \mathbb{Q}(\zeta_m)$](./latex/latex2png-ClassFieldTheory_76097731_-5.gif)
. Since
![$\mathbb{Q}(\zeta_m)\cap \mathbb{Q}(\zeta_{m'})=\mathbb{Q}(\zeta_{(m,m')})$](./latex/latex2png-ClassFieldTheory_86510772_-6.gif)
, the gcd of all such
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
's is the smallest
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
. If
![$p\nmid f$](./latex/latex2png-ClassFieldTheory_241783995_-5.gif)
, then
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
is unramified in
![$\mathbb{Q}(\zeta_f)\supseteq K$](./latex/latex2png-ClassFieldTheory_233792723_-5.gif)
. Conversely, if
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
is unramified in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
, we write
![$f=p^rf'$](./latex/latex2png-ClassFieldTheory_235664944_-4.gif)
, then the restriction map on inertia groups is
![$I(\mathbb{Q}(\zeta_{f})/p)\cong (\mathbb{Z}/p^r \mathbb{Z})^\times\rightarrow I(K/p)=\{1\}$](./latex/latex2png-ClassFieldTheory_181451156_-5.gif)
. Hence
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is contained in the fixed field of
![$\mathbb{Q}(\zeta_f)$](./latex/latex2png-ClassFieldTheory_99119596_-5.gif)
of
![$(\mathbb{Z}/p^r \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_24225209_-5.gif)
, i.e.,
![$\mathbb{Q}(\zeta_{f'})$](./latex/latex2png-ClassFieldTheory_236688841_-5.gif)
.
¡õ
Artin maps
Recall the following commutative diagram
We know that for
,
maps surjectively to the inertia group
and
maps surjectively to the wild inertia subgroup, i.e., the
-Sylow subgroup of
. For
, the element
(with 1 at the place
) maps to
, which is equal to
.
Let
![$K\subseteq \mathbb{Q}(\zeta_m)$](./latex/latex2png-ClassFieldTheory_76097731_-5.gif)
be a finite abelian extension
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
. Define the
Artin map ![$$\Psi_{K/\mathbb{Q}}: \mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times\twoheadrightarrow \Gal(K/\mathbb{Q})$$](./latex/latex2png-ClassFieldTheory_118086963_.gif)
by sending
![$\mathbb{R}_{>0}$](./latex/latex2png-ClassFieldTheory_180773804_-4.gif)
to
![$\{1\}$](./latex/latex2png-ClassFieldTheory_118978227_-5.gif)
and
![$\hat{\mathbb{Z}}^\times\rightarrow \Gal(K/\mathbb{Q})$](./latex/latex2png-ClassFieldTheory_214914286_-5.gif)
as the restriction
![$\Gal(\mathbb{Q}^\mathrm{cyc}/\mathbb{Q})\rightarrow \Gal(\mathbb{Q}(\zeta_m)/\mathbb{Q})\rightarrow \Gal(K/\mathbb{Q})$](./latex/latex2png-ClassFieldTheory_264433613_-5.gif)
. It is a continuous surjection. It follows that
![$\mathbb{Z}_p^\times$](./latex/latex2png-ClassFieldTheory_157244420_-7.gif)
maps surjectively to the inertia group
![$I(K/p)$](./latex/latex2png-ClassFieldTheory_61578446_-5.gif)
and
![$p\in \mathbb{Q}_p^\times$](./latex/latex2png-ClassFieldTheory_8928346_-7.gif)
maps to
![$\Frob_p^{-1}$](./latex/latex2png-ClassFieldTheory_189954079_-7.gif)
.
10/03/2012
![$\Psi_{K/\mathbb{Q}}(\mathbb{Q}_p^\times)=D(K/p)$](./latex/latex2png-ClassFieldTheory_60054028_-7.gif)
,
![$\Psi_{K/\mathbb{Q}}(\mathbb{Z}_p^\times)=I(K/p)$](./latex/latex2png-ClassFieldTheory_90940996_-7.gif)
. When
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
is unramified is
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
,
![$\Psi_{K/\mathbb{Q}}(u_p)=\Frob_p^{-1}$](./latex/latex2png-ClassFieldTheory_258075840_-7.gif)
(opposite to the usual Artin map), where
![$u_p$](./latex/latex2png-ClassFieldTheory_214273004_-5.gif)
is the image of
![$p\in \mathbb{Q}_p^\times\hookrightarrow \mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times$](./latex/latex2png-ClassFieldTheory_249579793_-7.gif)
.
We already know that
![$\Psi_{K/\mathbb{Q}}(\mathbb{Z}_p^\times)=I(K/p)$](./latex/latex2png-ClassFieldTheory_90940996_-7.gif)
. Without loss of generality we may assume
![$K=\mathbb{Q}(\zeta_m)$](./latex/latex2png-ClassFieldTheory_176656544_-5.gif)
. Suppose
![$p\nmid m$](./latex/latex2png-ClassFieldTheory_242242747_-5.gif)
is unramified, then
![$$u_p=(1,\ldots,1,p,1,\ldots,,1)=(1/p,\ldots,1/p,1,1/p,\ldots,1/p)\in \mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times$$](./latex/latex2png-ClassFieldTheory_13969342_.gif)
maps to
![$p^{-1}\mod{m}\in (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_78251555_-5.gif)
, i.e.,
![$\Frob_p^{-1}$](./latex/latex2png-ClassFieldTheory_189954079_-7.gif)
. For arbitrary
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
, we write
![$m=p^rm'$](./latex/latex2png-ClassFieldTheory_243006768_-4.gif)
. Then
![$\Psi_{\mathbb{Q}(\zeta_{m'})/\mathbb{Q}}(u_p)=p^{-1}\in (\mathbb{Z}/m' \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_33688825_-6.gif)
. Because any prime
![$\ell\ne p$](./latex/latex2png-ClassFieldTheory_32148658_-4.gif)
is unramified in
![$\mathbb{Q}(\zeta_{p^r})$](./latex/latex2png-ClassFieldTheory_159223708_-5.gif)
, we also know that
![$\Psi_{\mathbb{Q}(\zeta_{p^r})/\mathbb{Q}}(u_p)=1\in (\mathbb{Z}/p^r \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_80103581_-7.gif)
. Hence
![$$\Psi_{K/\mathbb{Q}}(u_p)=(p^{-1},1)\in (\mathbb{Z}/m' \mathbb{Z})^\times\times (\mathbb{Z}/p^r \mathbb{Z})^\times.$$](./latex/latex2png-ClassFieldTheory_81470598_.gif)
We know that
![$$\Psi(\mathbb{Q}_p^\times)=\Psi(\mathbb{Z}_p^\times)\langle\Psi(u_p)\rangle=(\mathbb{Z}/p^r \mathbb{Z})^\times\times \langle p\mod{m'}\rangle\cong D(K/\mathbb{Q}).$$](./latex/latex2png-ClassFieldTheory_8205109_.gif)
This completes the proof.
¡õ
Now for any
![$K\subseteq \mathbb{Q}^\mathrm{cyc}$](./latex/latex2png-ClassFieldTheory_84694294_-3.gif)
with infinite degree,
![$\Psi_{K/\mathbb{Q}}$](./latex/latex2png-ClassFieldTheory_125506446_-6.gif)
makes sense by taking the inverse limit over
![$K'\subseteq K$](./latex/latex2png-ClassFieldTheory_206464447_-3.gif)
finite over
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
.
![$\Psi_{K/\mathbb{Q}}$](./latex/latex2png-ClassFieldTheory_125506446_-6.gif)
is obviously continuous by construction. Since
![$\hat{\mathbb{Z}}^\times$](./latex/latex2png-ClassFieldTheory_186886302_0.gif)
is compact and the image is dense (surjective on every finite
![$\Gal(K/\mathbb{Q})$](./latex/latex2png-ClassFieldTheory_18108096_-5.gif)
, we find the
Artin map ![$\Psi_\mathbb{Q}:=\Psi_{\mathbb{Q}^\mathrm{cyc}/\mathbb{Q}}$](./latex/latex2png-ClassFieldTheory_57192453_-6.gif)
is surjective.
The following proposition summarizes easy properties of the Artin map
. We will see how they generalize for any global field.
is a bijection between the finite subextension of
and open subgroups (of finite index) of
. Indeed, any open subgroup
contains some
, hence
is a subgroup of
, hence corresponds to a finite extension
.
is continuous (and surjective onto
).
, the connected component of
.
Now let us turn to the local case
.
Suppose
, then
is unramified over
. Recall that
sending
to
, so
, where
is the order of
in
. On the other hand,
is totally ramified of degree
. Hence
is an isomorphism. Hence for general
, we have
and the inertia subgroup
.
Taking the inverse limit of the exact sequence
we obtain two isomorphic exact sequences ![$$\xymatrix{0\ar[r] &\I\ar[r] \ar[d]^\cong& \Gal(\mathbb{Q}_p^\mathrm{cyc}/\mathbb{Q}_p)\ar[r]\ar[d]^\cong &\Gal(\overline{\mathbb{F}_p}/\mathbb{F}_p)\ar[r]\ar[d]^\cong& 0 \\ 0\ar[r] & \mathbb{Z}_p^\times\ar[r] & \hat{\mathbb{Z}} \times\mathbb{Z}_p^\times\ar[r] & \hat{\mathbb{Z}}\ar[r] & 0.}$$](./latex/latex2png-ClassFieldTheory_34934691_.gif)
We define the
Artin map ![$\Psi_{\mathbb{Q}_p}: \mathbb{Q}_p^\times\cong q^\mathbb{Z}\times \mathbb{Z}_p^\times\rightarrow \Gal(\mathbb{Q}_p^\mathrm{cyc}/\mathbb{Q}_p)$](./latex/latex2png-ClassFieldTheory_216943650_-7.gif)
sending
![$$(p,x)\mapsto (-1, x)\in \mathbb{Z}\times \mathbb{Z}_p^\times\subseteq \hat{\mathbb{Z}}\times \mathbb{Z}_p^\times\cong\Gal(\mathbb{Q}_p^\mathrm{cyc}/\mathbb{Q}_p).$$](./latex/latex2png-ClassFieldTheory_5136286_.gif)
This is a continuous map with dense image and maps
![$\mathbb{Z}_p^\times$](./latex/latex2png-ClassFieldTheory_157244420_-7.gif)
surjective onto the inertia group.
10/05/2012
The following important proposition is left as an exercise.
(Local-global compatibility)
We have the following commutative diagram
Weil groups
Let
be a nonarchimedean local field (resp. a global function field) and
be the residue field (resp. the constant field). The residue field (resp. constant field) of
is
. Sow we have a continuous surjection
. Denote its kernel by
(which is the inertia group in the local case). As a group, the Weil group is simply
, i.e., the elements in
which induces integral powers of Frobenius on
. However, as we have seen in Remark 53,
may not be open in
under the subspace topology from
. But there exists a finer topology on
such that
is open in
. under this finer topology,
(in the local case) or
(in the global function field case) will be isomorphisms of topological groups.
More precisely, suppose we have a short exact sequence of profinite groups
and
.
There exists a unique topology on
![$W$](./latex/latex2png-ClassFieldTheory_43611154_-1.gif)
such that
![$I$](./latex/latex2png-ClassFieldTheory_42693650_0.gif)
is open in
![$W$](./latex/latex2png-ClassFieldTheory_43611154_-1.gif)
,
![$I$](./latex/latex2png-ClassFieldTheory_42693650_0.gif)
has the subspace topology under
![$I\hookrightarrow \Gamma$](./latex/latex2png-ClassFieldTheory_201823119_-1.gif)
and any splitting of
![$W\rightarrow \mathbb{Z}$](./latex/latex2png-ClassFieldTheory_38981706_-1.gif)
induces an isomorphism
![$W\cong I\times \mathbb{Z}$](./latex/latex2png-ClassFieldTheory_109992347_-1.gif)
of topological groups.
See the handouts for details.
¡õ
Under this topology on the Weil group
,
is continuous and has dense image (but is not a homeomorphism onto its image). Moreover, the topology is compatible under abelianization as in following proposition.
Let
![$\pi^\mathrm{ab}: \Gamma^\mathrm{ab}\rightarrow \hat{\mathbb{Z}}$](./latex/latex2png-ClassFieldTheory_111273207_-1.gif)
be the abelianization of
![$\pi$](./latex/latex2png-ClassFieldTheory_12809236_0.gif)
and
![$W'=(\pi^\mathrm{ab})^{-1}(\mathbb{Z})$](./latex/latex2png-ClassFieldTheory_98733216_-5.gif)
:
![$$\xymatrix{0\ar[r] & I\ar[r]\ar[d] & W\ar[r]\ar[d] & \mathbb{Z}\ar[r]\ar[d]^{\cong} &0\\ 0\ar[r] & I' \ar[r] & W' \ar[r]& \mathbb{Z}\ar[r] &0.}$$](./latex/latex2png-ClassFieldTheory_40965936_.gif)
Then
![$W'=W^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_229577769_-1.gif)
with
![$W\rightarrow W'$](./latex/latex2png-ClassFieldTheory_149680140_-1.gif)
the abelianization as topological groups (but not for
![$I\rightarrow I'$](./latex/latex2png-ClassFieldTheory_134996492_-1.gif)
).
The following theorem allows us to use Weil groups as a replacement of to classify finite abelian extensions of local or global function fields.
The maps
![$H\mapsto H\cap W$](./latex/latex2png-ClassFieldTheory_176923294_-1.gif)
and
![$\bar H' \mapsfrom H'$](./latex/latex2png-ClassFieldTheory_77418051_-1.gif)
gives a bijection between open subgroups of
![$\Gamma$](./latex/latex2png-ClassFieldTheory_108290532_0.gif)
and open finite index subgroup of
![$W$](./latex/latex2png-ClassFieldTheory_43611154_-1.gif)
. For any
![$H \subseteq\Gamma$](./latex/latex2png-ClassFieldTheory_188085277_-3.gif)
an open subgroup,
![$W/(W\cap H)$](./latex/latex2png-ClassFieldTheory_20364286_-5.gif)
is isomorphic to
![$\Gamma/H$](./latex/latex2png-ClassFieldTheory_207912683_-5.gif)
as discrete coset spaces.
This bijection extends to a bijection between closed subgroups
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
of
![$\Gamma$](./latex/latex2png-ClassFieldTheory_108290532_0.gif)
such that
![$\pi(H)=\{0\}$](./latex/latex2png-ClassFieldTheory_229367443_-5.gif)
or
![$\pi(H)$](./latex/latex2png-ClassFieldTheory_55861730_-5.gif)
has finite index in
![$\hat{\mathbb{Z}}$](./latex/latex2png-ClassFieldTheory_6257619_0.gif)
, and closed subgroups of
![$W$](./latex/latex2png-ClassFieldTheory_43611154_-1.gif)
. This will allows us to partially classify infinite abelian extensions of local fields and global function fields.
Statement of global class field theory
Property (a) in Theorem
21 uniquely characterizes
![$\Psi_K$](./latex/latex2png-ClassFieldTheory_146956762_-2.gif)
.
Observe that if
![$S$](./latex/latex2png-ClassFieldTheory_43349010_-1.gif)
is a finite set of places of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. Then
![$X_S=K^\times\{x\in \mathbb{A}_K^\times : x_v=1, \forall v\in S\}$](./latex/latex2png-ClassFieldTheory_22123309_-5.gif)
is dense in
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
by the weak approximation (Remark
47). If
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite abelian extension, we let
![$S$](./latex/latex2png-ClassFieldTheory_43349010_-1.gif)
contain
![$S_\infty$](./latex/latex2png-ClassFieldTheory_54331671_-2.gif)
and all the ramified places. Then
![$\Psi_{L/K}|_{X_S}$](./latex/latex2png-ClassFieldTheory_107572687_-6.gif)
is determined by (a), hence by continuity
![$\Psi_{L/K}$](./latex/latex2png-ClassFieldTheory_31691942_-6.gif)
is determined by (a).
¡õ
Assuming Theorem 21, let us prove the following "real version" of the existence theorem (and please hope for the "real real version").
(Existence Theorem)
![$L\mapsto \ker(\Psi_{L/K})$](./latex/latex2png-ClassFieldTheory_83032203_-6.gif)
is an inclusion reverse bijection between finite abelian extensions
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
and open subgroups of finite index in
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
.
It suffices to show the injectivity. Suppose
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
and
![$L'$](./latex/latex2png-ClassFieldTheory_149517330_0.gif)
are finite abelian extensions such that
![$\ker(\Psi_{L/K})=\ker(\Psi_{L'/K})$](./latex/latex2png-ClassFieldTheory_203344364_-6.gif)
. Let
![$H_L$](./latex/latex2png-ClassFieldTheory_238973972_-2.gif)
and
![$H_{L'}$](./latex/latex2png-ClassFieldTheory_20472586_-2.gif)
be the corresponding open subgroup in
![$G_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_227018301_-5.gif)
, then
![$\Psi_K^{-1}(H_L)=\Psi_K^{-1}(H_{L'})$](./latex/latex2png-ClassFieldTheory_106505286_-5.gif)
. Hence
![$H_L\cap \Im (\Psi_K)=H_{L'}\cap \Im(\Psi_K)$](./latex/latex2png-ClassFieldTheory_103460078_-5.gif)
. Suppose
![$g\in H_L\setminus H_{L'}$](./latex/latex2png-ClassFieldTheory_48474223_-5.gif)
, then there is an neighborhood
![$U(g)\subseteq H_L\setminus H_{L'}$](./latex/latex2png-ClassFieldTheory_105003555_-5.gif)
since
![$H_{L}$](./latex/latex2png-ClassFieldTheory_135755424_-2.gif)
and
![$H_{L'} $](./latex/latex2png-ClassFieldTheory_208994905_-2.gif)
are both open and closed. But
![$\Im(\Psi_K)$](./latex/latex2png-ClassFieldTheory_189573788_-5.gif)
is dense, this would contradict
![$H_L\cap \Im (\Psi_K)=H_{L'}\cap \Im(\Psi_K)$](./latex/latex2png-ClassFieldTheory_103460078_-5.gif)
. We conclude that
![$H_L=H_{L'}$](./latex/latex2png-ClassFieldTheory_128677366_-2.gif)
, thus
![$L=L'$](./latex/latex2png-ClassFieldTheory_32076859_0.gif)
.
¡õ
10/10/2012
Suppose
is a number field and
be the connected component of
. We have shown that
is profinite (Lemma 10).
is profinite, thus contains no divisible elements, hence
. But the image of
is dense, we know that
is surjective in this case. On the other hand,
is profinite implies that the intersection of all open subgroups
is trivial. We find that the intersection of all open subgroups (of finite index) of
is
, as any subgroup of finite index contains divisible elements
. Therefore
by the existence theorem. Namely, we have shown
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field, then
![$\Psi_K$](./latex/latex2png-ClassFieldTheory_146956762_-2.gif)
induces an isomorphism of topological groups
![$(\mathbb{A}_K^\times/K^\times)/D_K\cong G_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_209708460_-5.gif)
.
We thus deduce a stronger version of existence theorem for number fields.
The map
![$L\mapsto H=\ker(\Psi_{L/K})$](./latex/latex2png-ClassFieldTheory_144100529_-6.gif)
is a bijection between
all abelian extensions
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
and
closed subgroups of
![$H\subseteq \mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_236426981_-5.gif)
such that
![$H\supseteq (\prod_{v|\infty}K_v^\times)^0$](./latex/latex2png-ClassFieldTheory_205036412_-8.gif)
. Under this bijection,
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
is called the
class field of
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
.
Now suppose
is a global function field. We have an isomorphism of topological groups
, a product of a discrete group and a profinite group. Then the intersection of all open subgroup
is trivial, hence
is injective.
But
is not surjective, indeed we claim that if
, then
, an integral power of the Frobenius. In fact, for a finite extension
, then
is finite abelian and unramified everywhere and
(Theorem 13, or the base change of an etale map is etale). It suffices to show that
. For
,
lifts
, By definition,
as
is unramified everywhere and
. In other words, we have proved the following diagram commutes ![$$\xymatrix{0\ar[r] & (\mathbb{A}_K^\times)^1/K^\times \ar[r] \ar[d] & \mathbb{A}_K^\times/K^\times \ar[r] \ar[d]^{\Psi_K} & q^\mathbb{Z} \ar[r] \ar[d] & 0\\ 0 \ar[r] & I_K \ar[r] & G_K^\mathrm{ab} \ar[r] & G_{\bar k}\cong \hat{\mathbb{Z}} \ar[r] & 0.}$$](./latex/latex2png-ClassFieldTheory_77237112_.gif)
It follows that
. We claim that
is actually an isomorphism. We already know it is injective, so it suffices to show the surjectivity. Choose
such that
. Then there exists a unique continuous homomorphism
sending 1 to
by the universal property of profinite groups. Hence
as topological groups. Under this identification,
is simply
. Let
. Then
is a closed subgroup and
, hence the closure of
in
is
. Since the image of
is dense, it follows that
, which proves the surjectivity.
We have proved
![$\Psi_K$](./latex/latex2png-ClassFieldTheory_146956762_-2.gif)
induces vertical isomorphisms of topological groups
![$$\xymatrix{0 \ar[r] & (\mathbb{A}_K^\times)^1/K^\times \ar[r] \ar[d]^{\cong} & \mathbb{A}_K^\times/K^\times \ar[r] \ar[d]^{\cong}& q^\mathbb{Z} \ar[r] \ar[d]^{\cong} & 0\\0\ar[r] & I_K \ar[r] & W \ar[r] & \mathbb{Z} \ar[r] &0.}$$](./latex/latex2png-ClassFieldTheory_107890666_.gif)
So
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
can viewed as the Weil group
![$W$](./latex/latex2png-ClassFieldTheory_43611154_-1.gif)
and
![$G_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_227018301_-5.gif)
is the profinite completion of
![$W$](./latex/latex2png-ClassFieldTheory_43611154_-1.gif)
.
![$L\mapsto \ker(\Psi_{L/K})$](./latex/latex2png-ClassFieldTheory_83032203_-6.gif)
gives a bijection between
all abelian extensions
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
such that the constant field extension
![$l/k$](./latex/latex2png-ClassFieldTheory_120943636_-5.gif)
is either finite or equal to
![$\bar k$](./latex/latex2png-ClassFieldTheory_62546417_0.gif)
, and
closed subgroups of
![$\mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_14481591_-5.gif)
.
10/12/2012
Norm and Verlagerung functoriality
Let
be a global field and
be any finite separable extension.
gives us a canonical map
. We also have the compatibility of local and global norms: ![$$\xymatrix{\mathbb{A}_L^\times\ar[r]^-{N_{L/K}} &\mathbb{A}_K^\times\\ L_w^\times \ar[r]^-{N_{L_w/K_v}} \ar[u] & K_v^\times. \ar[u]}$$](./latex/latex2png-ClassFieldTheory_109197814_.gif)
(Norm functoriality)
The follows diagram commutes:
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is finite abelian, then
![$\Psi_{L/K}N_{L/K}=1$](./latex/latex2png-ClassFieldTheory_183971975_-6.gif)
. In other words,
![$N_{L/K}(\mathbb{A}_L^\times/L^\times)\subseteq\ker(\Psi_{L/K})$](./latex/latex2png-ClassFieldTheory_96380224_-6.gif)
.
Take
![$K'{}=L$](./latex/latex2png-ClassFieldTheory_153001198_0.gif)
in the previous theorem.
¡õ
(Global norm index inequality)
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite and separable extension. Then
![$[\mathbb{A}_K^\times/K^\times: N_{L/K}(\mathbb{A}_L^\times/L^\times)]\le[L:K]$](./latex/latex2png-ClassFieldTheory_34235028_-6.gif)
.
We will give an easy analytic proof later (cf. Exercise
17). In fact, the cohomological proof will even give a division relation.
¡õ
(Existence theorem III)
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite and separable extension. Then
![$[\mathbb{A}_K^\times/K^\times: N_{L/K}(\mathbb{A}_L^\times/L^\times)]=[L:K]$](./latex/latex2png-ClassFieldTheory_250450648_-6.gif)
if and only if
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian, in which case
![$\ker(\Psi_{L/K})=N_{L/K}(\mathbb{A}_L^\times/L^\times)$](./latex/latex2png-ClassFieldTheory_161766553_-6.gif)
.
It remains to prove the "only if" direction, which is left as an exercise (hint: the left-hand-side is always
![$[L^\mathrm{ab}:K]$](./latex/latex2png-ClassFieldTheory_234995990_-5.gif)
.
¡õ
Now let us briefly turn to the verlagerung functoriality of the Artin map.
Let
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
be a group and
![$S\subseteq G$](./latex/latex2png-ClassFieldTheory_206201691_-3.gif)
be a subgroup of finite index. For any
![$g,\sigma\in G$](./latex/latex2png-ClassFieldTheory_113332348_-4.gif)
, let
![$f_\sigma$](./latex/latex2png-ClassFieldTheory_206325471_-4.gif)
be the smallest
![$f\ge1$](./latex/latex2png-ClassFieldTheory_50558614_-4.gif)
such that
![$\sigma g^f \sigma^{-1}\in S$](./latex/latex2png-ClassFieldTheory_91831604_-4.gif)
. This only depends the choice of
![$\sigma$](./latex/latex2png-ClassFieldTheory_208953823_0.gif)
in
![$S\setminus G/ \langle g\rangle$](./latex/latex2png-ClassFieldTheory_141340296_-5.gif)
, a finite double coset. Write
![$\{\sigma_1,\ldots,\sigma_r\}$](./latex/latex2png-ClassFieldTheory_11101157_-5.gif)
be the coset representatives. We define the
verlagerung (or
transfer)
![$\Ver: g\mapsto \prod_{i=1}^r\sigma_i g^{f_{\sigma_i}}\sigma_i^{-1}\in S$](./latex/latex2png-ClassFieldTheory_9029857_-5.gif)
. Then
![$\Ver: G^\mathrm{ab}\rightarrow S^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_31247569_-1.gif)
is a group homomorphism. In terms of group cohomology, this is the restriction map
![$\Res: H_1(G, \mathbb{Z})\rightarrow H_1(S, \mathbb{Z})$](./latex/latex2png-ClassFieldTheory_185553504_-5.gif)
and functorial in
![$(G,S)$](./latex/latex2png-ClassFieldTheory_22771326_-5.gif)
.
Suppose
is a field and
is finite separable. Then
is of finite index (depending on the choice of an isomorphisms
). We then obtain the verlagerung
, a continuous map of the topological abelianization of
and
, which does not depend on the choice of
.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite separable extension. The following diagram commutes:
Suppose
![$K'/K$](./latex/latex2png-ClassFieldTheory_96635846_-5.gif)
is finite Galois and
![$L=K'L$](./latex/latex2png-ClassFieldTheory_37849438_0.gif)
. It suffices to show at the finite level. The remaining check will be an easy calculation which we leave as an exercise.
¡õ
Statement of local class field theory
10/15/2012
Analogously, the local existence theorem will follow from the local norm index inequality.
(Local norm index inequality)
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a local field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite and separable extension. Then
![$[K^\times: N_{L/K}(L^\times)]\le[L:K]$](./latex/latex2png-ClassFieldTheory_210016358_-6.gif)
.
(Existence Theorem)
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a local field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite and separable extension. Then
![$[K^\times: N_{L/K}(L^\times)]=[L:K]$](./latex/latex2png-ClassFieldTheory_142352851_-6.gif)
if and only if
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian, in which case,
![$N_{L/K}(L^\times)\cong\ker(\Psi_{L/K})$](./latex/latex2png-ClassFieldTheory_218739718_-6.gif)
.
The local-global compatibility will follow from defining the global Artin map via "gluing" local Artin maps.
(Local-global compatibility)
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global field and
![$v\in V_K$](./latex/latex2png-ClassFieldTheory_60378006_-2.gif)
. Then we have a commutative diagram
![$$\xymatrix{ K_v^\times \ar[d] \ar[r]^-{\Psi_{K_v}} & G_{K_v}^\mathrm{ab}\ar[d] \\ \mathbb{A}_K^\times/K^\times \ar[r]^-{\Psi_K} & G_K^\mathrm{ab} .}$$](./latex/latex2png-ClassFieldTheory_66566876_.gif)
If
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is finite abelian and
![$w|v$](./latex/latex2png-ClassFieldTheory_191859692_-5.gif)
, then
![$L_w/K_v$](./latex/latex2png-ClassFieldTheory_21169748_-5.gif)
is also finite abelian and we have a corresponding commutative diagram at finite level.
We can then derive part of (a) in global class field theory (Theorem 21) using the local-global compatibility at archimedean places.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field and
![$v\in V_K$](./latex/latex2png-ClassFieldTheory_60378006_-2.gif)
is a real place. If
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is finite abelian, then
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is unramified in
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
if and only if
![$\Psi_{L/k}$](./latex/latex2png-ClassFieldTheory_40080550_-6.gif)
kills all
![$K_v^\times$](./latex/latex2png-ClassFieldTheory_36812834_-4.gif)
. Otherwise
![$\Psi_{L/K}$](./latex/latex2png-ClassFieldTheory_31691942_-6.gif)
kills
![$(K_v^\times)_{>0}$](./latex/latex2png-ClassFieldTheory_69051978_-5.gif)
and
![$\Psi_{L/K}(-1)$](./latex/latex2png-ClassFieldTheory_126063150_-6.gif)
is the complex conjugation.
Ray class fields and conductors
Now let us discuss the classical formulation of class field theory in terms of ideal classes.
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field. A
modulus of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a formal product
![$\mathfrak{m}=\prod_{v\in V_K} \mathfrak{p}_v^{e_v}$](./latex/latex2png-ClassFieldTheory_208532144_-7.gif)
, where
![$e_v\ge0$](./latex/latex2png-ClassFieldTheory_50657110_-3.gif)
and
![$e_v=0$](./latex/latex2png-ClassFieldTheory_33715861_-2.gif)
for almost all
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
; for
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
a real place,
![$e_v=0,1$](./latex/latex2png-ClassFieldTheory_49674068_-4.gif)
;
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
a complex place,
![$e_v=0$](./latex/latex2png-ClassFieldTheory_33715861_-2.gif)
. We denote by
![$\mathfrak{m}_v=\mathfrak{p}_v^{e_v}$](./latex/latex2png-ClassFieldTheory_33836166_-4.gif)
the
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
-component of
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
. We write
![$v|\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530711_-5.gif)
or
![$\mathfrak{p}_v|\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_160657242_-5.gif)
if
![$e_v>0$](./latex/latex2png-ClassFieldTheory_34764437_-2.gif)
. The modulus is used to keep track of the ramification of the places of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
in some sense.
Let
![$x\in \mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_259462054_-5.gif)
, we say
![$x\equiv1 \mod{\mathfrak{m}}$](./latex/latex2png-ClassFieldTheory_189319970_-1.gif)
if and only if
![$x_v\equiv1\mod{\mathfrak{m}_v}$](./latex/latex2png-ClassFieldTheory_37035158_-2.gif)
for any
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
. This is a multiplicative condition, so it makes sense to say that
![$x\equiv y\mod{\mathfrak{m}}$](./latex/latex2png-ClassFieldTheory_172542755_-4.gif)
whenever
![$x_vy_v^{-1}\equiv 1\mod{\mathfrak{m}_v}$](./latex/latex2png-ClassFieldTheory_130141184_-4.gif)
.
For
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
, each modulus is of the form
![$\mathfrak{m}=m$](./latex/latex2png-ClassFieldTheory_156924834_-1.gif)
or
![$\mathfrak{m}=m\infty$](./latex/latex2png-ClassFieldTheory_207826401_-1.gif)
, where
![$m\in \mathbb{Z}_{>0}$](./latex/latex2png-ClassFieldTheory_93283453_-4.gif)
. It follows easily that an idele
![$x=(x_f,x_\infty)\equiv 1\mod{m}$](./latex/latex2png-ClassFieldTheory_204288562_-5.gif)
if and only the finite part
![$x_f\in \ker(\hat{\mathbb{Z}}^\times\rightarrow(\mathbb{Z}/m \mathbb{Z})^\times)$](./latex/latex2png-ClassFieldTheory_194506141_-5.gif)
(which can be thought of the usual congruence relation
![$x\equiv 1\mod{m}$](./latex/latex2png-ClassFieldTheory_232015051_0.gif)
), and
![$x\equiv1\mod{m\infty}$](./latex/latex2png-ClassFieldTheory_241207410_0.gif)
if we further have
![$x_\infty>0$](./latex/latex2png-ClassFieldTheory_25724491_-2.gif)
.
Suppose
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is a modulus, we define
![$U_\mathfrak{m}=\{x\in \mathbb{A}_K^\times: x\equiv 1\mod{\mathfrak{m}} \}$](./latex/latex2png-ClassFieldTheory_60395842_-5.gif)
, namely
![$$U_\mathfrak{m}=\prod_{v|\infty\atop e_v=0}K_v^\times\prod_{v|\infty\atop e_v=1}(K_v^\times)_{>0}\prod_{v\nmid\infty\atop e_v=0}\mathcal{O}_v^\times\prod_{v\nmid\infty\atop e_v>0}(1+\mathfrak{p}_v^{e_v}).$$](./latex/latex2png-ClassFieldTheory_241985250_.gif)
It is an open subgroup of
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
and
![$U_{\mathfrak{m}'}\subseteq U_{\mathfrak{m}}$](./latex/latex2png-ClassFieldTheory_208144905_-3.gif)
if and only if
![$\mathfrak{m}|\mathfrak{m}'$](./latex/latex2png-ClassFieldTheory_267506482_-5.gif)
. We know that
![$\{U_\mathfrak{m}\}$](./latex/latex2png-ClassFieldTheory_177011038_-5.gif)
forms a cofinal filtered system of open subgroups in
![$\mathbb{A}_K^\times$](./latex/latex2png-ClassFieldTheory_262173692_-5.gif)
. This generalizes Proposition
4.
For
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
and
![$\mathfrak{m}=m\infty$](./latex/latex2png-ClassFieldTheory_207826401_-1.gif)
, we have
![$I_K^{(\mathfrak{m})}=\{a/b \in \mathbb{Q}^\times_{>0}: (a,m)=(b,m)=1\}$](./latex/latex2png-ClassFieldTheory_228433537_-6.gif)
. We have a surjection
![$I_K^{(\mathfrak{m})}\rightarrow (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_35488914_-5.gif)
and the kernel is exactly
![$P_{\mathfrak{m}}=\{x\in \mathbb{Q}_{>0}: x\equiv 1\mod{m}\}$](./latex/latex2png-ClassFieldTheory_124359517_-5.gif)
. Therefore
![$\Cl_{\mathfrak{m}}(\mathbb{Q})\cong (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_112578904_-5.gif)
. Similarly, when
![$\mathfrak{m}=m$](./latex/latex2png-ClassFieldTheory_156924834_-1.gif)
, we find that
![$\Cl_\mathfrak{m}(\mathbb{Q})= (\mathbb{Z}/m \mathbb{Z})^\times/\pm1$](./latex/latex2png-ClassFieldTheory_228134004_-5.gif)
.
10/17/2012
There are isomorphisms
We define
![$\mathbb{A}_{K,\mathfrak{m}}^\times=\{ x\in \mathbb{A}_K^\times : x\equiv1 \mod{\mathfrak{m}}\}$](./latex/latex2png-ClassFieldTheory_73728499_-8.gif)
(it properly contains
![$U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530588_-3.gif)
) and
![$K_{\mathfrak{m}}^\times=K^\times\cap \mathbb{A}_{K,\mathfrak{m}}^\times$](./latex/latex2png-ClassFieldTheory_161788795_-8.gif)
. Then we have an injection
![$$\mathbb{A}_{K,\mathfrak{m}}^\times/K^\times_{\mathfrak{m}}U_\mathfrak{m}\hookrightarrow\mathbb{A}_K^\times/K^\times U_\mathfrak{m}.$$](./latex/latex2png-ClassFieldTheory_263721728_.gif)
It is actually an isomorphism by weak approximation. Now define the homomorphism
![$$\mathbb{A}_{K,\mathfrak{m}}^\times\rightarrow I_K^{(\mathfrak{m})},\quad x\mapsto \prod_{v\nmid\infty} \mathfrak{p}_v^{\ord_v(x_v)},$$](./latex/latex2png-ClassFieldTheory_36260256_.gif)
which is obviously a surjection with kernel
![$U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530588_-3.gif)
. Taking quotient by the image of
![$K_\mathfrak{m}^\times$](./latex/latex2png-ClassFieldTheory_36056801_-5.gif)
, we obtain that
![$\Cl_\mathfrak{m}(K)\cong \mathbb{A}_{K,\mathfrak{m}}^\times/K_{\mathfrak{m}}^\times U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_257129307_-8.gif)
.
¡õ
The
ray class field of the modulus
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is the class field
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
corresponding to
![$K^\times U_\mathfrak{m}/K^\times\subseteq \mathbb{A}_K^\times/K^\times$](./latex/latex2png-ClassFieldTheory_137365354_-5.gif)
. When
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field,
![$K_\mathfrak{m}/K$](./latex/latex2png-ClassFieldTheory_145319586_-5.gif)
is finite abelian. When
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global function field,
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
contains
![$K\otimes_k\bar k$](./latex/latex2png-ClassFieldTheory_175129136_-2.gif)
since the image of
![$K^\times U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_165929499_-3.gif)
has trivial image when projected to
![$q^\mathbb{Z} $](./latex/latex2png-ClassFieldTheory_268404096_-4.gif)
.
When
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
, we have
![$\mathbb{Q}_{m\infty}=\mathbb{Q}(\zeta_m)$](./latex/latex2png-ClassFieldTheory_117165150_-5.gif)
and
![$\mathbb{Q}_{m}=\mathbb{Q}(\zeta_m+\bar \zeta_m)$](./latex/latex2png-ClassFieldTheory_263084613_-5.gif)
and the following diagram commutes
We summarize the easy properties of ray class fields as follows.
is unramified at all
.
![$K_\mathfrak{m} \cap K_{\mathfrak{m}'}=K_{\gcd(\mathfrak{m} ,\mathfrak{m}')}$](./latex/latex2png-ClassFieldTheory_31789160_-6.gif)
- If
, then
.
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite abelian extension. We say a modulus
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is
admissible for
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
if
![$L\subseteq K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_85687315_-3.gif)
. The gcd of all admissible modulus of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is called the
conductor ![$\mathfrak{f}_{L/K}$](./latex/latex2png-ClassFieldTheory_73953380_-6.gif)
of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
. It follows that
![$L\subseteq K_{\mathfrak{f}_{L/K}}$](./latex/latex2png-ClassFieldTheory_29560350_-7.gif)
.
When
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
.
![$\mathfrak{f}_{L/K}$](./latex/latex2png-ClassFieldTheory_73953380_-6.gif)
is essentially the same as Proposition
3, except that
![$\infty\nmid \mathfrak{f}$](./latex/latex2png-ClassFieldTheory_122442104_-5.gif)
if and only if
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
is totally real.
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is admissible for
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
if and only if
![$N_{L/K}(\mathbb{A}_L^\times)\supseteq U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_218774629_-6.gif)
.
The "if" direction follows from the fact that
![$K^\times U_\mathfrak{m}/K^\times\subseteq N_{L/K}(\mathbb{A}_L^\times/L^\times)$](./latex/latex2png-ClassFieldTheory_84460593_-6.gif)
. For the "only if" direction, suppose
![$L\subseteq K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_85687315_-3.gif)
, then the local-global compatibility implies that
![$\Psi_{L_w/K_v}(1+\mathfrak{p}_v^{e_v})=1$](./latex/latex2png-ClassFieldTheory_176748180_-6.gif)
for
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
nonarchimedean (and a similar thing for archimedean places), hence by the definition of idelic norm,
![$N_{L/K}(\mathbb{A}_L^\times)\supseteq N_{L_w/K_v}(L_w^\times)\supseteq 1+\mathfrak{p}_v^{e_v}$](./latex/latex2png-ClassFieldTheory_124085278_-6.gif)
. Therefore
![$U_\mathfrak{m}\subseteq N_{L/K}(\mathbb{A}_L^\times)$](./latex/latex2png-ClassFieldTheory_201310398_-6.gif)
.
¡õ
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is ramified in
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
if and only if
![$v| \mathfrak{f}_{L/K}$](./latex/latex2png-ClassFieldTheory_141061584_-6.gif)
.
Let
![$N=N_{L/K}(\mathbb{A}_L^\times/L^\times)\supseteq K^\times U_\mathfrak{m}/K^\times$](./latex/latex2png-ClassFieldTheory_61662073_-6.gif)
, where
![$\mathfrak{m}=\mathfrak{f}_{L/K}$](./latex/latex2png-ClassFieldTheory_147327141_-6.gif)
. If
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is ramified, the
![$\mathcal{O}_v^\times\not\subseteq N$](./latex/latex2png-ClassFieldTheory_88294385_-4.gif)
, hence
![$\mathcal{O}_v^\times\not\subseteq K^\times U_\mathfrak{m}/K^\times$](./latex/latex2png-ClassFieldTheory_227612197_-5.gif)
, hence
![$\mathcal{O}_v^\times\not\subseteq U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_262184087_-4.gif)
, which implies that
![$v|\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530711_-5.gif)
. Now suppose
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
is unramified, then for any
![$w|v$](./latex/latex2png-ClassFieldTheory_191859692_-5.gif)
,
![$N_{L_w/K_v}(L_w^\times)\supseteq\mathcal{O}_v^\times$](./latex/latex2png-ClassFieldTheory_14261194_-6.gif)
. Then we can find a modulus
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
such that
![$v\nmid \mathfrak{m}$](./latex/latex2png-ClassFieldTheory_76696536_-5.gif)
and
![$N_{L/K}(\mathbb{A}_L^\times)\supseteq U_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_218774629_-6.gif)
. It follows that
![$v|\mathfrak{f}_{L/K}$](./latex/latex2png-ClassFieldTheory_194482118_-6.gif)
by the previous lemma and the definition of
![$\mathfrak{f}_{L/K}$](./latex/latex2png-ClassFieldTheory_73953380_-6.gif)
.
¡õ
Ideal-theoretic formulation of global class field theory
Let
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
be a finite abelian extension and
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
be a modulus which is divisible by all ramified places. We define the
Artin symbol
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is admissible if and only if
![$P_\mathfrak{m}\subseteq \ker \Art_{L/K}$](./latex/latex2png-ClassFieldTheory_48270307_-6.gif)
. (This is the way Artin originally introduced the notion of admissible moduli. The existence of admissible moduli is truly surprising and is the key difficulty of class field theory!)
10/19/2012
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite extension and
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is a modulus of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We denote by
![$I_L^{(\mathfrak{m})}$](./latex/latex2png-ClassFieldTheory_62017589_-5.gif)
the free abelian group generated by prime ideals of
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
whose restriction to
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
are coprime to
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
. The usual ideal norm restricts to a group homomorphism
![$N_{L/K}: I_L^{(\mathfrak{m})}\rightarrow I_K^{(\mathfrak{m})}$](./latex/latex2png-ClassFieldTheory_262016780_-6.gif)
. We denote the image of
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
by
![$N_{L/K}(\mathfrak{m})$](./latex/latex2png-ClassFieldTheory_260854689_-6.gif)
.
Let
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
be a finite extension. Suppose
![$K'/K$](./latex/latex2png-ClassFieldTheory_96635846_-5.gif)
is finite abelian and
![$L'{}=K'L$](./latex/latex2png-ClassFieldTheory_35168101_0.gif)
. Then we have a commutative diagram
![$$\xymatrix{I_L^{(\mathfrak{n})} \ar[r] \ar[d]^{N_{L/K}}& \Gal(L'/L)\ar[d] \\ I_K^{(\mathfrak{m})} \ar[r] & \Gal(K'/K),}$$](./latex/latex2png-ClassFieldTheory_65915879_.gif)
where
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is a modulus of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
divisible by the primes of ramified in
![$K'$](./latex/latex2png-ClassFieldTheory_148468754_0.gif)
,
![$\mathfrak{n}$](./latex/latex2png-ClassFieldTheory_258492647_-1.gif)
is a modulus of
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
divisible by primes ramified in
![$L'$](./latex/latex2png-ClassFieldTheory_149517330_0.gif)
or restrict to primes of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
ramified in
![$K'$](./latex/latex2png-ClassFieldTheory_148468754_0.gif)
.
Weber L-functions
Before stepping into the cohomological proof of class field theory, we will discuss various applications of class field theory in the following several weeks. From now on we will assume
is a number field for simplicity (though some results are also valid for function fields). Write the degree
.
We defined the
Dedekind zeta function ![$$\zeta_K(s)=\prod_{\mathfrak{p}}\frac{1}{1- (N \mathfrak{p})^{-s}}=\sum_{\mathfrak{a}}\frac{1}{(N\mathfrak{a})^s},$$](./latex/latex2png-ClassFieldTheory_72838360_.gif)
where
![$N\mathfrak{a}=\#(\mathcal{O}_K/\mathfrak{a})$](./latex/latex2png-ClassFieldTheory_240483173_-5.gif)
is the absolute norm.
We omit the proof of the simple analytic property.
![$\zeta_K(s)$](./latex/latex2png-ClassFieldTheory_116495366_-5.gif)
is analytic on
![$\Re(s)>1-1/N$](./latex/latex2png-ClassFieldTheory_96644784_-5.gif)
except a simple pole at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
.
Let
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
be a modulus of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
and
![$\mathcal{R}\in \Cl_\mathfrak{m}(K)$](./latex/latex2png-ClassFieldTheory_85208497_-5.gif)
be an ideal class. We define
A similar analytic property holds for
too.
![$\zeta_K(s,\mathcal{R})$](./latex/latex2png-ClassFieldTheory_263210525_-5.gif)
is analytic on
![$\Re(s)>1-1/N$](./latex/latex2png-ClassFieldTheory_96644784_-5.gif)
except a simple pole at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
. The residue at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
depends on the modulus
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
but not on ideal class
![$\mathcal{R}$](./latex/latex2png-ClassFieldTheory_117082110_-1.gif)
.
Recall that for
a finite abelian group, we have the notion of Pontryakin dual
consisting of characters of
, and the following elementary properties holds:
canonically.
- If
, then ![$\sum_{g\in G}\chi(g)=
\begin{cases}
\# G, & \chi=1,\\
0, & \chi\ne1.
\end{cases}$](./latex/latex2png-ClassFieldTheory_253532261_-26.gif)
- If
, then ![$\sum_{\chi\in \hat G}\chi(g)=
\begin{cases}
\# G, & g=1,\\
0, & g\ne1.
\end{cases}$](./latex/latex2png-ClassFieldTheory_139693486_-26.gif)
The
Weber
-function for a character
![$\chi: \Cl_\mathfrak{m}(K)\rightarrow \mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_193333625_-5.gif)
is defined to be
![$$L_{K,\mathfrak{m}}(s,\chi)=\prod_{\mathfrak{p}\nmid \mathfrak{m}}\frac{1}{1-\chi(\mathfrak{p})(N \mathfrak{p})^{-s}}=\sum_{(\mathfrak{m},\mathfrak{a})=1}\frac{\chi(\mathfrak{a})}{(N \mathfrak{a})^s}=\sum_{\mathcal{R}\in \Cl_\mathfrak{m}(K)}\chi(\mathcal{R})\zeta_K(s,\mathcal{R}).$$](./latex/latex2png-ClassFieldTheory_87762983_.gif)
It follows from the previous theorem that
![$L_{K,\mathfrak{m}}(s,\chi)$](./latex/latex2png-ClassFieldTheory_101445163_-5.gif)
is analytic on
![$\Re(s)>1-1/N$](./latex/latex2png-ClassFieldTheory_96644784_-5.gif)
except a possible simple pole at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
. When
![$\chi\ne1$](./latex/latex2png-ClassFieldTheory_170750809_-4.gif)
, it is actually analytic at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
by the previous proposition, since all the residues of
![$\zeta_K(s,\mathcal{R})$](./latex/latex2png-ClassFieldTheory_263210525_-5.gif)
at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
are the same.
Consider
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
and
![$\mathfrak{m}=m\infty$](./latex/latex2png-ClassFieldTheory_207826401_-1.gif)
. Then
![$$\zeta_\mathbb{Q}(s)=\sum_{n\ge1}\frac{1}{n^s}$$](./latex/latex2png-ClassFieldTheory_22200420_.gif)
is simply the
Riemann zeta function. A character
![$\chi: \Cl_\mathfrak{m}(\mathbb{Q})\rightarrow \mathbb{C}$](./latex/latex2png-ClassFieldTheory_247185955_-5.gif)
is the same thing as a
Dirichlet character ![$(\mathbb{Z}/m \mathbb{Z})^\times\rightarrow \mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_86537882_-5.gif)
. Then
![$$L_{\mathbb{Q},m\infty}=\sum_{n\ge1}\frac{\chi(n)}{n^s}$$](./latex/latex2png-ClassFieldTheory_109149688_.gif)
is simply a
Dirichlet
-function, where we extend
![$\chi$](./latex/latex2png-ClassFieldTheory_12356549_-4.gif)
on
![$\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_50353232_0.gif)
by letting
![$\chi(x)=0$](./latex/latex2png-ClassFieldTheory_262301775_-5.gif)
whenever
![$(x,m)\ne1$](./latex/latex2png-ClassFieldTheory_208708753_-5.gif)
.
Now class field theory easily imply the following result on special values of Weber
-functions.
![$L_{K,\mathfrak{m}}(1,\chi)\ne0$](./latex/latex2png-ClassFieldTheory_169581089_-5.gif)
if
![$\chi\ne1$](./latex/latex2png-ClassFieldTheory_170750809_-4.gif)
.
By global class field theory, there exists a class field
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
such that
![$\Cl_\mathfrak{m}(K)\cong\Gal(K_\mathfrak{m}/K)$](./latex/latex2png-ClassFieldTheory_244911043_-5.gif)
. So
![$\chi$](./latex/latex2png-ClassFieldTheory_12356549_-4.gif)
can be viewed as a character of
![$G_K$](./latex/latex2png-ClassFieldTheory_222131220_-2.gif)
. Let
![$\mathfrak{n}$](./latex/latex2png-ClassFieldTheory_258492647_-1.gif)
be a modulus of
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
divisible exactly by the primes of
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
restricting to
![$\mathfrak{p}|\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_16827037_-5.gif)
. Then as a special case of the lemma below (
![$L=K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_256853756_-3.gif)
)
![$$L_{K_\mathfrak{m},\mathfrak{n}}(s,1)=\prod_{\chi}L_{K,\mathfrak{m}}(s,\chi).$$](./latex/latex2png-ClassFieldTheory_148894074_.gif)
The result then follows since each of the
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
-functions
![$L_{K_\mathfrak{m},\mathfrak{n}}(s,1)$](./latex/latex2png-ClassFieldTheory_125515779_-5.gif)
and
![$L_{K,\mathfrak{m}}(s,1)$](./latex/latex2png-ClassFieldTheory_20016871_-5.gif)
has a simple pole at
![$s=1$](./latex/latex2png-ClassFieldTheory_251884564_0.gif)
.
¡õ
10/22/2012
Let
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
be a finite abelian extension. Suppose
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is an admissible modulus for
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
and
![$\mathfrak{n}$](./latex/latex2png-ClassFieldTheory_258492647_-1.gif)
is exactly divisible by primes of
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
restricting to primes dividing
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
. Then
![$$L_{L,\mathfrak{n}}(s,1)=\prod_\chi L_{K, \mathfrak{m}}(\chi),$$](./latex/latex2png-ClassFieldTheory_51468961_.gif)
where
![$\chi$](./latex/latex2png-ClassFieldTheory_12356549_-4.gif)
runs over all characters
![$\Cl_\mathfrak{m}(K)\twoheadrightarrow\Gal(L/K)\xrightarrow{\chi}\mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_172463977_-5.gif)
.
This equality actually holds at the level of local Euler factors. Say
![$\mathfrak{p}\nmid \mathfrak{m}$](./latex/latex2png-ClassFieldTheory_198529894_-5.gif)
, then
![$\mathfrak{p}$](./latex/latex2png-ClassFieldTheory_256395495_-4.gif)
is unramified and
![$\mathfrak{p} \mathcal{O}_L=\prod_i \mathfrak{q}_i$](./latex/latex2png-ClassFieldTheory_190393085_-5.gif)
, we claim that
![$$\prod_i \frac{1}{1-(N \mathfrak{q}_i)^{-s}}=\prod_\chi \frac{1}{1-\chi(\mathfrak{p})(N \mathfrak{p})^{-s}}.$$](./latex/latex2png-ClassFieldTheory_22555629_.gif)
Write
![$n=[L:K]$](./latex/latex2png-ClassFieldTheory_15598015_-5.gif)
, then
![$N \mathfrak{q}_{i=1}^{n/f}=N \mathfrak{p}^f$](./latex/latex2png-ClassFieldTheory_37033943_-5.gif)
. The right hand side becomes
![$$\frac{1}{(1-(N \mathfrak{p})^{-sf})^{n/f}}.$$](./latex/latex2png-ClassFieldTheory_161098354_.gif)
Write
![$y=(N \mathfrak{p})^{-s}$](./latex/latex2png-ClassFieldTheory_94028101_-5.gif)
for short. Taking the logarithms of both sides, it reduces to show that
![$$\frac{n}{f}\sum_{m\ge1} \frac{y^{fm}}{m}=\sum_\chi\sum_{m\ge1}\frac{\chi(\mathfrak{p})^my^m}{m}.$$](./latex/latex2png-ClassFieldTheory_220198321_.gif)
Notice that
![$\chi(\mathfrak{p})=\chi(\Frob_\mathfrak{p})$](./latex/latex2png-ClassFieldTheory_70806802_-5.gif)
and
![$\Frob_\mathfrak{p}$](./latex/latex2png-ClassFieldTheory_87907852_-5.gif)
has order
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
in
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
. Then the character
![$\chi\mapsto \chi(\mathfrak{p})^m$](./latex/latex2png-ClassFieldTheory_195046650_-5.gif)
of the Pontryagin dual of
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
is nontrivial if and only if
![$f\nmid m$](./latex/latex2png-ClassFieldTheory_242201787_-5.gif)
. Therefore the right-hand-side is simply
![$$\sum_\chi\sum_{f\nmid m}\frac{y^m}{m}=n\sum_{m\ge1}\frac{y^{fm}}{fm}=\frac{n}{f}\sum_{m\ge1}\frac{y^{\mathfrak{m}}}{m},$$](./latex/latex2png-ClassFieldTheory_202667947_.gif)
which coincides with the left-hand-side.
¡õ
Consider a one-dimensional Galois representation
![$\rho: G_K\rightarrow \mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_166769204_-4.gif)
. Then
![$\rho$](./latex/latex2png-ClassFieldTheory_239694908_-4.gif)
factors through a finite cyclic extension
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
, where
![$L=(\bar K)^{\ker\rho}$](./latex/latex2png-ClassFieldTheory_162151184_-5.gif)
. Let
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
be the conductor of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
, then
![$\rho$](./latex/latex2png-ClassFieldTheory_239694908_-4.gif)
induces a character
![$\Cl_\mathfrak{m}(K)\twoheadrightarrow\Gal(L/K)\xrightarrow{\chi} \mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_118611264_-5.gif)
. By definition,
![$$L_{K,\mathfrak{m}}(s,\chi)=L(s,\rho),$$](./latex/latex2png-ClassFieldTheory_187696380_.gif)
since
![$\mathbb{C}^{I_\mathfrak{p}}\ne0$](./latex/latex2png-ClassFieldTheory_82994942_-4.gif)
if and only if
![$\mathfrak{p}$](./latex/latex2png-ClassFieldTheory_256395495_-4.gif)
is unramified in
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
. In other words, Weber
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
-functions can be viewed as the same thing as one-dimensional Artin
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
-functions via
class field theory. Notice Weber
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
-functions involves geometry of numbers (ideal classes, etc.), which are crucial for establishing the analytic properties. On the other hand, the functoriality properties of Artin
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
-functions give handy ways to establish non-vanishing results of special values. This picture motivates the Langlands program of the study of general Artin
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
-functions via relating Galois representations and automorphic representations and class field theory can be viewed as the Langlands program for
![$GL_1$](./latex/latex2png-ClassFieldTheory_35877946_-2.gif)
.
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a finite abelian extension with
![$G=\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_15982207_-5.gif)
. Then
![$\Ind_{G_L}^{G_K}(\mathbf{1}_L)\cong\bigoplus_\chi \chi$](./latex/latex2png-ClassFieldTheory_210210107_-8.gif)
(notice the left-hand-side is the regular representation of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
). Thus we have
![$$\zeta_L(s)=L(s,\mathbf{1}_L)=L(s,\Ind_{G_L}^{G_K}\mathbf{1}_L)=L(s,\bigoplus_\chi \chi)=\prod_\chi L(s,\chi).$$](./latex/latex2png-ClassFieldTheory_171179310_.gif)
This is analogous to the previous lemma except that more local Euler factors are involved here.
General Artin
-functions can be reduced to one-dimensional Artin
-functions via Brauer's induction theorem.
(Brauer)
Suppose
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is a finite group and
![$\rho: G\rightarrow GL(V)\cong GL_n(\mathbb{C})$](./latex/latex2png-ClassFieldTheory_20008226_-5.gif)
. Then there exists
![$n_1,\ldots, n_r\in \mathbb{Z}$](./latex/latex2png-ClassFieldTheory_23926690_-4.gif)
,
![$H_1,\ldots, H_i$](./latex/latex2png-ClassFieldTheory_246858516_-4.gif)
subgroups of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
and characters
![$\chi_i: H_i\rightarrow \mathbb{C}^\times$](./latex/latex2png-ClassFieldTheory_199584051_-4.gif)
such that
![$$\rho\cong\sum_{i=1}^r n_i\Ind_{H_i}^G\chi_i,$$](./latex/latex2png-ClassFieldTheory_214790410_.gif)
as virtual representations.
With the same notation,
Chebotarev's density theorem
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a number field of degree
![$N $](./latex/latex2png-ClassFieldTheory_151155730_0.gif)
. Let
![$P $](./latex/latex2png-ClassFieldTheory_153252882_0.gif)
be a set of finite primes of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. We define the
natural density of
![$P $](./latex/latex2png-ClassFieldTheory_153252882_0.gif)
to be the limit (if it exists)
![$$\mu(P)=\lim_{x\rightarrow\infty}\frac{\# \{\mathfrak{p} \in P: N \mathfrak{p}<x\}}{\#\{\mathfrak{p}: N \mathfrak{p}<x\}}.$$](./latex/latex2png-ClassFieldTheory_164387677_.gif)
We define the
Dirichlet density to be
10/24/2012
The following basic properties of the (Dirichlet) density
follows easily from definition.
- If
has a density, then
.
- If
is finite, then
.
- If
is the disjoint union of
and
and two of
,
,
have density, then so does the third one and
.
- If
and both have density, then
.
- If
has a density and
,then
.
- If
has density and
is the complement of
, then
has density and
.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field. Denote the
degree of a prime
![$\mathfrak{p}$](./latex/latex2png-ClassFieldTheory_256395495_-4.gif)
by
![$\deg \mathfrak{p}=[\mathcal{O}_K/\mathfrak{p}: \mathbb{F}_p]$](./latex/latex2png-ClassFieldTheory_265746067_-5.gif)
. Prove that
![$\{\mathfrak{p} : \deg \mathfrak{p}=1\}$](./latex/latex2png-ClassFieldTheory_71127519_-5.gif)
has density 1. (Not to be confused with the density of of split primes of
![$\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_40916048_-3.gif)
in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
, which will be shown to be
![$1/N $](./latex/latex2png-ClassFieldTheory_67269689_-5.gif)
).
Let
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
be a modulus and
![$\mathcal{R}_0\in \Cl_\mathfrak{m}(K)$](./latex/latex2png-ClassFieldTheory_233423754_-5.gif)
. Then
![$\delta(\{\mathfrak{p}\in \mathcal{R}_0\})=1/\#\Cl_\mathfrak{m}(K)$](./latex/latex2png-ClassFieldTheory_84491194_-5.gif)
.
Notice that
![$$\log L_{K,\mathfrak{m}}(s,\chi)\sim_1\sum_{\mathfrak{p}\nmid \mathfrak{m}}\frac{\chi(\mathfrak{p})}{(N \mathfrak{p})^{s}}=\sum_{\mathcal{R}\in \Cl_\mathfrak{m}(K)}\chi(\mathcal{R})\sum_{\mathfrak{p}\in \mathcal{R}}\frac{1}{(N \mathfrak{p})^s}.$$](./latex/latex2png-ClassFieldTheory_233930529_.gif)
On the other hand, since
![$L_{K,\mathfrak{m}}(s,\chi)$](./latex/latex2png-ClassFieldTheory_101445163_-5.gif)
is analytic at 1 whenever
![$\chi\ne1$](./latex/latex2png-ClassFieldTheory_170750809_-4.gif)
, we know that
![$$\log(s-1)^{-1}\sim_1\log\zeta_K(s)\sim_1\log L_{K,\mathfrak{m}}(s,1)\sim_1\sum_\chi \chi(\mathcal{R}_0)^{-1}\log L_{K,\mathfrak{m}}(s,\chi).$$](./latex/latex2png-ClassFieldTheory_155375358_.gif)
Expanding the last sum gives
![$$\sum_{\chi,\mathcal{R}}\chi(\mathcal{R}/\mathcal{R}_0)\sum_{\mathfrak{p}\in \mathcal{R}}\frac{1}{(N \mathfrak{p})^s}=\#\Cl_\mathfrak{m}(K)\sum_{\mathfrak{p}\in \mathcal{R}_0}\frac{1}{(N \mathfrak{p})^s}.$$](./latex/latex2png-ClassFieldTheory_250795540_.gif)
The desired result then follows.
¡õ
When
![$K=\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_191911032_-3.gif)
,
![$\mathfrak{m}=m\cdot\infty$](./latex/latex2png-ClassFieldTheory_58539693_-1.gif)
, we have
![$\Cl_\mathfrak{m}(\mathbb{Q})\cong (\mathbb{Z}/m \mathbb{Z})^\times$](./latex/latex2png-ClassFieldTheory_138262522_-5.gif)
and an ideal class is given by simply given by a residue class modulo
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
. From the previous proposition we recover the classical theorem of Dirichlet on primes in progressions:
![$\delta(\{p:p\equiv n\mod{m}\})=1/\varphi(m)$](./latex/latex2png-ClassFieldTheory_117283616_-5.gif)
. Namely, the primes are equidistributed modulo
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
.
(Chebotarev's density)
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is a Galois (but not necessarily abelian) extension of global fields with
![$G=\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_15982207_-5.gif)
and
![$N=[L:K]$](./latex/latex2png-ClassFieldTheory_15595967_-5.gif)
. Let
![$\sigma\in G$](./latex/latex2png-ClassFieldTheory_113332988_-1.gif)
and
![$C $](./latex/latex2png-ClassFieldTheory_139621394_-1.gif)
be the conjugacy class of
![$\sigma$](./latex/latex2png-ClassFieldTheory_208953823_0.gif)
in
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
(with
![$c=\#C$](./latex/latex2png-ClassFieldTheory_242579089_-4.gif)
). Then
![$S_C=\{\mathfrak{p}: \mathfrak{p} \text{ unramified in } L, \Frob_\mathfrak{p}\in C\}$](./latex/latex2png-ClassFieldTheory_100480343_-5.gif)
has density
![$\delta(S_C)=c/N$](./latex/latex2png-ClassFieldTheory_33511568_-5.gif)
. In other words, the Frobenius conjugacy classes are equidistributed in
![$\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_159637122_-5.gif)
.
Let us first show the case that
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian. Let
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
be the conductor of
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
. Let
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
be the kernel of
![$\Art_{L/K}: I_K^{(\mathfrak{m})}\rightarrow G$](./latex/latex2png-ClassFieldTheory_184203254_-6.gif)
. Then
![$C=\{\sigma\}$](./latex/latex2png-ClassFieldTheory_36733838_-5.gif)
and
![$S_C=\{\mathfrak{p}\nmid \mathfrak{m}: \Frob_\mathfrak{p}=\sigma\}=\{\mathfrak{p} \nmid \mathfrak{m}: \mathfrak{p}\in \mathfrak{a}+H\}$](./latex/latex2png-ClassFieldTheory_25868579_-5.gif)
, where
![$\mathfrak{a}$](./latex/latex2png-ClassFieldTheory_264746777_-1.gif)
is any preimage of
![$\sigma$](./latex/latex2png-ClassFieldTheory_208953823_0.gif)
. Since
![$(\mathfrak{a}+H)/P_\mathfrak{m}\subseteq \Cl_\mathfrak{m}(K)$](./latex/latex2png-ClassFieldTheory_261638636_-5.gif)
contains exactly
![$[H: P_\mathfrak{m}]$](./latex/latex2png-ClassFieldTheory_141555773_-5.gif)
ideal classes, we know that
![$$\delta(S_C)=\frac{[H: P_\mathfrak{m}]}{\#\Cl_\mathfrak{m}(K)}=\frac{[H: P_\mathfrak{m}]}{[I_K^{(\mathfrak{m})}: P_\mathfrak{m}]}=\frac{1}{[I_K^{(\mathfrak{m})}:H]}=\frac{1}{N}$$](./latex/latex2png-ClassFieldTheory_62303515_.gif)
as needed. This is the only step where the usage of class field theory is crucial.
10/26/2012
For the general case, we let
be the fixed field of
under the cyclic group generated by
. Then
is a cyclic extension with Galois group
and we can reduce the previous case as follows. Let
be the primes
of
unramified in
and
. By the abelian case, we know that
. Also, let
be the primes
in
such that
, then
by Exercise 15. We claim that for any
,
. Assuming this claim, we know that
as desired.
It remains to show the claim. Let
be the set of primes
of
such that
is unramified in
and
. For
, write
and
. Since
acts trivially on
, it also acts trivially on
. Therefore
, i.e.
. Since
, we know that
is the unique prime of
over
. On the other hand, given
, let
and
be a prime of
over
. Since
has order
, we know that
. Hence
is the unique prime of
over
and
. Thus
. In this way we have exhibited a bijection between
and
.
Now let
and
a prime of
over
. We can choose
. Then
is the orbit of
under the centralizer
of
. So
which proves the claim.
¡õ
Split primes
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
be a finite separable extension. We define
![$$\Spl(L/K)=\{v\in V_K: v\text{ splits completely in } L\}.$$](./latex/latex2png-ClassFieldTheory_66480105_.gif)
For
![$S\subseteq V_K$](./latex/latex2png-ClassFieldTheory_47399490_-3.gif)
, we also define
![$\Spl_S(L/K)=\Spl(L/K)\setminus S$](./latex/latex2png-ClassFieldTheory_201826888_-5.gif)
.
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is
Galois and
![$\delta(S)=0$](./latex/latex2png-ClassFieldTheory_50068527_-5.gif)
. Then
![$\Spl(L/K)$](./latex/latex2png-ClassFieldTheory_159584130_-5.gif)
and
![$\Spl_S(L/K)$](./latex/latex2png-ClassFieldTheory_178775951_-5.gif)
have density
![$1/[L:K]$](./latex/latex2png-ClassFieldTheory_15603615_-5.gif)
.
Use Chebotarev's Density Theorem
34 for
![$\sigma=1$](./latex/latex2png-ClassFieldTheory_221086565_0.gif)
.
¡õ
- Suppose
are finite separable. Then
.
- Suppose
is finite separable and
is its Galois closure. Then
. In particular,
if and only if
is Galois.
If
![$\Spl(L/K)$](./latex/latex2png-ClassFieldTheory_159584130_-5.gif)
has density 1, then
![$L=K$](./latex/latex2png-ClassFieldTheory_266505196_0.gif)
.
Apply the previous exercise to the Galois closure of
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
.
¡õ
We can now prove the following results without using class field theory.
for
.
- (Global norm index inequality)
for
a finite extension of number fields.
Suppose
![$L,L'/K$](./latex/latex2png-ClassFieldTheory_171810050_-5.gif)
are finite
Galois,
![$\delta(S)=0$](./latex/latex2png-ClassFieldTheory_50068527_-5.gif)
. Then the following are equivalent:
.
.
.
(a) implies (c) and (c) implies (b) are obvious. For (b) implies (a), notice that
![$\Spl_S(LL'/K)=\Spl_S(L)\cap\Spl_S(L')=\Spl_S(L'/K)$](./latex/latex2png-ClassFieldTheory_45929240_-5.gif)
. Therefore
![$LL'{}=L'$](./latex/latex2png-ClassFieldTheory_233667878_0.gif)
and
![$L\subseteq L'$](./latex/latex2png-ClassFieldTheory_83388581_-3.gif)
.
¡õ
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
and
![$L'/K$](./latex/latex2png-ClassFieldTheory_171799610_-5.gif)
are finite Galois,
![$\delta(S)=0$](./latex/latex2png-ClassFieldTheory_50068527_-5.gif)
. Then
![$L=L'$](./latex/latex2png-ClassFieldTheory_32076859_0.gif)
if and only if
![$\Spl_S(L/K)=\Spl_S(L'/K)$](./latex/latex2png-ClassFieldTheory_27201406_-5.gif)
. In other words, a Galois extension is determined by the set of split primes.
Hilbert class fields
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a number field. The
Hilbert class field ![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is defined to be the ray class field
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
for
![$\mathfrak{m}=1$](./latex/latex2png-ClassFieldTheory_155614114_-1.gif)
. Then
![$\Gal(H/K)=\Cl(K)$](./latex/latex2png-ClassFieldTheory_129805840_-5.gif)
and
![$H/K$](./latex/latex2png-ClassFieldTheory_188576788_-5.gif)
is unramified at every place (including archimedean places). The
narrow Hilbert class field ![$H^+$](./latex/latex2png-ClassFieldTheory_235762708_0.gif)
of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is defined to be the ray class field
![$K_\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259530748_-3.gif)
, where
![$\mathfrak{m}$](./latex/latex2png-ClassFieldTheory_259541223_-1.gif)
is the product of all real places of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. Then
![$\Gal(H^+/K)\cong\Cl^+(K)$](./latex/latex2png-ClassFieldTheory_112772217_-5.gif)
(the
narrow class group) and
![$H^+/K$](./latex/latex2png-ClassFieldTheory_238909088_-5.gif)
is unramified at all finite places. Notice that
![$H\subseteq H^+$](./latex/latex2png-ClassFieldTheory_251444494_-3.gif)
and
![$\Cl^+(K)$](./latex/latex2png-ClassFieldTheory_49972302_-5.gif)
surjectis onto
![$\Cl(K)$](./latex/latex2png-ClassFieldTheory_215719449_-5.gif)
.
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
(resp.
![$H^+$](./latex/latex2png-ClassFieldTheory_235762708_0.gif)
) is the maximal abelian extension of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
which is unramified everywhere (resp. at all finite places).
If
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is unramified everywhere, then
![$\mathfrak{f}_{L/K}=1$](./latex/latex2png-ClassFieldTheory_115409139_-6.gif)
by Proposition
12. Hence
![$L\subseteq K_{1}=H$](./latex/latex2png-ClassFieldTheory_229194169_-3.gif)
. Similarly for
![$H^+$](./latex/latex2png-ClassFieldTheory_235762708_0.gif)
.
¡õ
10/31/2012
Suppose
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is finite extension of number fields and is totally ramified at some
![$v\in V_K$](./latex/latex2png-ClassFieldTheory_60378006_-2.gif)
. Then
![$h_K\mid h_L$](./latex/latex2png-ClassFieldTheory_209681762_-5.gif)
.
Since
![$L H_K/L$](./latex/latex2png-ClassFieldTheory_29293984_-5.gif)
is abelian and unramified everywhere, we know that
![$L H_K\subseteq H_L$](./latex/latex2png-ClassFieldTheory_133007298_-3.gif)
. So it suffices to show that
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
and
![$H_L$](./latex/latex2png-ClassFieldTheory_238973972_-2.gif)
are linearly disjoint over
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
(i.e.
![$L\otimes_K H_K\cong LH_K$](./latex/latex2png-ClassFieldTheory_143100981_-2.gif)
. Suppose
![$H_K=K(\alpha)$](./latex/latex2png-ClassFieldTheory_47797881_-5.gif)
with
![$f\in K[x]$](./latex/latex2png-ClassFieldTheory_219973225_-5.gif)
the minimal polynomial of
![$\alpha$](./latex/latex2png-ClassFieldTheory_205233679_0.gif)
. Let
![$g\in L[x]$](./latex/latex2png-ClassFieldTheory_48527767_-5.gif)
be a monic polynomial dividing
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
over
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
. Since
![$H_K/K$](./latex/latex2png-ClassFieldTheory_29526368_-5.gif)
is Galois,
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
splits into linear factors over
![$H_K$](./latex/latex2png-ClassFieldTheory_238908436_-2.gif)
, which implies
![$g$](./latex/latex2png-ClassFieldTheory_42038290_-4.gif)
also splits over
![$H_K$](./latex/latex2png-ClassFieldTheory_238908436_-2.gif)
. Hence
![$g\in H_K[x]\cap L[x]$](./latex/latex2png-ClassFieldTheory_152588256_-5.gif)
, which is equal to
![$K[x]$](./latex/latex2png-ClassFieldTheory_262179781_-5.gif)
by the assumption that
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is totally ramified at some place. Hence
![$g\in K[x]$](./latex/latex2png-ClassFieldTheory_219907689_-5.gif)
and
![$f$](./latex/latex2png-ClassFieldTheory_41972754_-4.gif)
is irreducible over
![$L$](./latex/latex2png-ClassFieldTheory_42890258_0.gif)
as needed.
¡õ
Consider
![$K=\mathbb{Q}(\zeta_m+\overline{\zeta_m})$](./latex/latex2png-ClassFieldTheory_261611327_-5.gif)
,
![$L=\mathbb{Q}(\zeta_m)$](./latex/latex2png-ClassFieldTheory_176656545_-5.gif)
. We know that
![$h_K\mid h_L$](./latex/latex2png-ClassFieldTheory_209681762_-5.gif)
using the previous proposition. When
![$m=p$](./latex/latex2png-ClassFieldTheory_152728596_-4.gif)
is a prime, the famous criterion of Kummer asserts that
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
divides
![$h_L/h_K$](./latex/latex2png-ClassFieldTheory_163579311_-5.gif)
(indeed equivalent to that
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
divides
![$h_L$](./latex/latex2png-ClassFieldTheory_104756244_-2.gif)
) if and only if
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
divides the numerator of some Bernoulli number
![$B_2,B_4,\cdots, B_{p-3}$](./latex/latex2png-ClassFieldTheory_108043153_-5.gif)
. Such a prime
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
is called
irregular.
Artin's principal ideal theorem
(Principal Ideal Theorem)
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a number field and
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
be its Hilbert class field. Then any ideal
![$\mathfrak{a}\subseteq \mathcal{O}_K$](./latex/latex2png-ClassFieldTheory_22950460_-3.gif)
becomes principal in
![$\mathcal{O}_H$](./latex/latex2png-ClassFieldTheory_113351606_-2.gif)
.
Using class field theory, it will reduce to the following purely group-theoretic theorem (we omit the proof).
(Furtwangler)
Let
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
be a finite group and
![$H=[G,G]$](./latex/latex2png-ClassFieldTheory_13762625_-5.gif)
. Then
![$\Ver: G^\mathrm{ab}\rightarrow H^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_31247580_-1.gif)
is trivial.
(Proof of the Principal Ideal Theorem)
Let
![$H'$](./latex/latex2png-ClassFieldTheory_145323026_0.gif)
be the Hilbert class field of
![$H=H_K$](./latex/latex2png-ClassFieldTheory_29526370_-2.gif)
. Then
![$H'/K$](./latex/latex2png-ClassFieldTheory_171799610_-5.gif)
is Galois since
![$H' $](./latex/latex2png-ClassFieldTheory_177370132_0.gif)
is intrinsic to
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. Notice that
![$H $](./latex/latex2png-ClassFieldTheory_144864274_0.gif)
is the maximal abelian subextension of
![$H' $](./latex/latex2png-ClassFieldTheory_177370132_0.gif)
. Therefore
![$\Gal(H'/H)=[G,G]$](./latex/latex2png-ClassFieldTheory_194849816_-5.gif)
and
![$\Gal(H/K)=G^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_96733497_-5.gif)
, where
![$G=\Gal(H'/K)$](./latex/latex2png-ClassFieldTheory_152679675_-5.gif)
. We have the following commutative diagram
![$$\xymatrix{I_K \ar[r]^-{\Art_{H/K}} \ar[d] & \Gal(H/K)=\Gal(H'/K)^\mathrm{ab} \ar@<6ex>[d]^{\Ver} \\ I_H \ar[r]^-{\Art_{H'/H}} & \Gal(H'/H)=\Gal(H'/H)^\mathrm{ab}}$$](./latex/latex2png-ClassFieldTheory_175320404_.gif)
By the previous theorem, we know that the
![$\Art_{H'/H}(\mathfrak{a}\mathcal{O}_H)=0$](./latex/latex2png-ClassFieldTheory_58427379_-6.gif)
, hence
![$\mathfrak{a}\mathcal{O}_H$](./latex/latex2png-ClassFieldTheory_163012191_-2.gif)
is principal.
¡õ
Class field towers
The principal ideal theorem motivates the following construction. Let
be a number field and
be the Hilbert class field of
. The we obtain the Golod-Shafarevich tower
Does
stabilize (i.e.,
)? The answer in general is no.
The analysis on this tower may be easier if we consider a single prime
at one time. Let
be the maximal unramified abelian
-extension of
. We obtain a tower
We now state a theorem of Golod-Shafarevich (for the proofs, cf. Cassels-Frolich Ch. IX).
(Golod-Shafarevich)
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a number field of degree
![$N $](./latex/latex2png-ClassFieldTheory_151155730_0.gif)
. If
![$[K_\infty^{(p)}:K]<\infty$](./latex/latex2png-ClassFieldTheory_239349528_-5.gif)
, then
![$$d^{(p)}\Cl(K)<2+2\sqrt{N+1},$$](./latex/latex2png-ClassFieldTheory_93482350_.gif)
where
![$d^{(p)}(G):=\dim_{\mathbb{F}_p}G/pG$](./latex/latex2png-ClassFieldTheory_64598239_-6.gif)
is the
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
-rank of a finite group of
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
.
On the other hand, Brumer's theorem provides a lower bound on
.
(Brumer)
Suppose
![$K/\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_42969992_-5.gif)
is Galois of degree
![$N $](./latex/latex2png-ClassFieldTheory_151155730_0.gif)
. Let
![$t_k^{(p)}$](./latex/latex2png-ClassFieldTheory_226078023_-5.gif)
be the number of primes
![$q $](./latex/latex2png-ClassFieldTheory_145912850_-4.gif)
such that
![$p\mid e(\mathfrak{q}/q)$](./latex/latex2png-ClassFieldTheory_164906201_-5.gif)
for any
![$\mathfrak{q}$](./latex/latex2png-ClassFieldTheory_255346919_-4.gif)
above
![$q $](./latex/latex2png-ClassFieldTheory_145912850_-4.gif)
. Then
Suppose
![$K/\mathbb{Q}$](./latex/latex2png-ClassFieldTheory_42969992_-5.gif)
is Galois of degree
![$N $](./latex/latex2png-ClassFieldTheory_151155730_0.gif)
. If
![$t_K^{(p)}\ge2N+2\sqrt{N+1}$](./latex/latex2png-ClassFieldTheory_239834871_-5.gif)
, then
![$[K_\infty^{(p)}: K]=\infty$](./latex/latex2png-ClassFieldTheory_182006816_-5.gif)
.
When
![$N=p=2$](./latex/latex2png-ClassFieldTheory_66816348_-4.gif)
, it follows from the previous corollary that a quadratic field has infinite Golod-Shafarevich tower whenever the number of ramified primes is at least 8, e.g.,
![$\mathbb{Q}(\sqrt{d})$](./latex/latex2png-ClassFieldTheory_37531685_-5.gif)
where
![$d$](./latex/latex2png-ClassFieldTheory_41841682_0.gif)
has at least 8 different prime factors. In particular, there are infinite such quadratic fields.
Hilbert class fields of global function fields
Now suppose
is a global function field. Then
is the maximal abelian unramified extension of
. In particular,
and
is infinite. To get better situation, we may ask what is the maximal unramified extension of
with constant field
. As class fields with constant fields
corresponds to subgroups
that surjects onto
under
, we know that a maximal unramified extension with constant field
corresponds to a minimal subgroup
such that there exists an element
with
. However, there may be more than one such minimal subgroup. The set of such subgroups
forms a principal homogeneous space under
(in geometrical terms, it is simply
). So there are
such groups
and hence there are
maximal abelian unramified extension of
with constant field
.
11/02/2012
Nevertheless, the following construction gives a maximal unramified extension which is canonical in some sense. Choose
with
. Then
modulo
is independent on the choice of
and the subgroup
is canonically defined. Let
be the class field of
. We then have an exact sequence
So
is everywhere unramified of degree
with constant field the degree
extension of
.
We end the discussion by applying a similar idea to prove a useful proposition concerning
-adic characters of the Weil group of a global function field.
Let
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
be a Hausdorff group. A continuous character
![$\chi: W_K\rightarrow G$](./latex/latex2png-ClassFieldTheory_189034512_-4.gif)
is called
unramified if the class field corresponding to
![$\ker\chi$](./latex/latex2png-ClassFieldTheory_9602255_-4.gif)
is unramified at
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a global function field of characteristic
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
and
![$E $](./latex/latex2png-ClassFieldTheory_141718546_0.gif)
is a finite extension of
![$\mathbb{Q}_\ell$](./latex/latex2png-ClassFieldTheory_90741589_-3.gif)
for some
![$\ell\ne p$](./latex/latex2png-ClassFieldTheory_32148658_-4.gif)
. Let
![$\chi: W_K\rightarrow\mathcal{O}_E^\times$](./latex/latex2png-ClassFieldTheory_28141034_-5.gif)
be a continuous homomorphism unramified outside a finite set
![$S$](./latex/latex2png-ClassFieldTheory_43349010_-1.gif)
. Then there exists
![$c\in \mathcal{O}_E^\times$](./latex/latex2png-ClassFieldTheory_154299521_-5.gif)
and
![$\chi_1: W_K\rightarrow\mathcal{O}_E^\times$](./latex/latex2png-ClassFieldTheory_193113547_-5.gif)
continuous of finite order such that
![$\chi=\chi_1\cdot c^{\deg(\cdot)}$](./latex/latex2png-ClassFieldTheory_2750736_-4.gif)
.
Since
![$\mathcal{O}_E^\times$](./latex/latex2png-ClassFieldTheory_112289573_-5.gif)
is abelian,
![$\chi$](./latex/latex2png-ClassFieldTheory_12356549_-4.gif)
factors through
![$W_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_227022397_-5.gif)
. Pick
![$\sigma\in W_K^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_8309622_-5.gif)
such that
![$\deg \sigma=1$](./latex/latex2png-ClassFieldTheory_236553733_-4.gif)
. Let
![$c=\chi(\sigma)$](./latex/latex2png-ClassFieldTheory_134251270_-5.gif)
and
![$\chi_1=\chi\cdot c^{-\deg(\cdot)}$](./latex/latex2png-ClassFieldTheory_8168151_-4.gif)
. Notice that
![$\chi_1(W_K)=\chi_1(I)$](./latex/latex2png-ClassFieldTheory_241019128_-5.gif)
by construction. But
![$I\cong (\mathbb{A}_K^\times)^1/K^\times$](./latex/latex2png-ClassFieldTheory_60373118_-5.gif)
, it suffices to show that
![$\chi(K^\times U_{(1)}/K^\times)$](./latex/latex2png-ClassFieldTheory_191976680_-6.gif)
is finite as
![$\Pic^0(X)$](./latex/latex2png-ClassFieldTheory_263407325_-5.gif)
is finite. But
![$K^\times U_{(1)}/K^\times\cong U_{(1)}/K^\times\cap U_{(1)}\cong U_{(1)}/k^\times$](./latex/latex2png-ClassFieldTheory_29493605_-6.gif)
and by assumption
![$\chi(\mathcal{O}_v^\times)=1$](./latex/latex2png-ClassFieldTheory_74838695_-5.gif)
for
![$v\not\in S$](./latex/latex2png-ClassFieldTheory_26615858_-4.gif)
, it suffices to show that
![$\chi(\prod_{v\in S}\mathcal{O}_v^\times)$](./latex/latex2png-ClassFieldTheory_208973036_-7.gif)
is finite. But
![$\mathcal{O}_E^\times$](./latex/latex2png-ClassFieldTheory_112289573_-5.gif)
has a finite index pro-
![$\ell$](./latex/latex2png-ClassFieldTheory_25588796_0.gif)
group and
![$\prod_{v\in S}\mathcal{O}_v^\times$](./latex/latex2png-ClassFieldTheory_188454997_-7.gif)
has a finite index pro-
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
group. Because there is no nontrivial map from a pro-
![$p$](./latex/latex2png-ClassFieldTheory_42628114_-4.gif)
group to a pro-
![$\ell$](./latex/latex2png-ClassFieldTheory_25588796_0.gif)
group, the image must be finite.
¡õ
Grunwald-Wang theorem
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field. Suppose
![$\Char(K)\nmid n$](./latex/latex2png-ClassFieldTheory_186189103_-5.gif)
and
![$K^\times$](./latex/latex2png-ClassFieldTheory_112818142_0.gif)
contains the
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th roots of unity. If
![$x\in K$](./latex/latex2png-ClassFieldTheory_129131150_-1.gif)
is an
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th power in
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
for almost all
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
, then
![$x$](./latex/latex2png-ClassFieldTheory_43152402_0.gif)
itself is an
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th power in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
Let
![$\alpha\in K^\mathrm{sep}$](./latex/latex2png-ClassFieldTheory_102283942_-1.gif)
be an
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th root of
![$x$](./latex/latex2png-ClassFieldTheory_43152402_0.gif)
and
![$f(t)\in K[t]$](./latex/latex2png-ClassFieldTheory_49382977_-5.gif)
be its minimal polynomial. So
![$f(t)\mid t^n-x$](./latex/latex2png-ClassFieldTheory_131339663_-5.gif)
. If
![$x$](./latex/latex2png-ClassFieldTheory_43152402_0.gif)
is an
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th power in
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
, then
![$f(t)$](./latex/latex2png-ClassFieldTheory_216757306_-5.gif)
splits completely in
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
as
![$K^\times$](./latex/latex2png-ClassFieldTheory_112818142_0.gif)
contains the
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th roots of unity. Since
![$f(t)$](./latex/latex2png-ClassFieldTheory_216757306_-5.gif)
is separable, it follows that
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
splits completely in
![$K_v(\alpha)$](./latex/latex2png-ClassFieldTheory_184790035_-5.gif)
. Now the split primes
![$\Spl(K(\alpha)/K)$](./latex/latex2png-ClassFieldTheory_75136186_-5.gif)
has density 1, hence
![$K(\alpha)=K$](./latex/latex2png-ClassFieldTheory_184372440_-5.gif)
.
¡õ
- 16 is an 8-th power in
and in
for
, but not in
(hence not in
, which is obvious).
- Let
, then 16 is an 8-th power in
for any place
but not an 8-th power in
.
(Grunwald-Wang)
Let
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
be a global field. If
![$x\in K$](./latex/latex2png-ClassFieldTheory_129131150_-1.gif)
is a
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th power in
![$K_v$](./latex/latex2png-ClassFieldTheory_247434220_-2.gif)
for almost all
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
, then
![$x$](./latex/latex2png-ClassFieldTheory_43152402_0.gif)
is an
![$n$](./latex/latex2png-ClassFieldTheory_42497042_0.gif)
-th power in
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
, except potentially if
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field and
![$K(\zeta_{2^t})/K$](./latex/latex2png-ClassFieldTheory_61162919_-5.gif)
is not cyclic, where
![$n=2^t\cdot m$](./latex/latex2png-ClassFieldTheory_20278229_0.gif)
with
![$m$](./latex/latex2png-ClassFieldTheory_42431506_0.gif)
odd.
The proof only uses the result on split primes (Corollary 10), so it is not really an application of class field theory.
11/05/2012
It suffices to treat the case that
![$n=p^a$](./latex/latex2png-ClassFieldTheory_31754596_-4.gif)
is a power of a prime. In fact, suppose
![$n=n_1n_2$](./latex/latex2png-ClassFieldTheory_62057004_-2.gif)
with
![$n_1, n_2$](./latex/latex2png-ClassFieldTheory_206386768_-4.gif)
coprime, then by the Euclidean algorithm we can write
![$a_1n_1+a_2n_2=1$](./latex/latex2png-ClassFieldTheory_138732963_-2.gif)
. Suppose
![$x=y_1^{n_1}=y_2^{n_2}$](./latex/latex2png-ClassFieldTheory_221915488_-4.gif)
, then
![$x=x^{a_1n_1+a_2n_2}=y_2^{a_1 n} y_1^{a_2 n}=(y_1^{a_1}y_2^{a_2})^n$](./latex/latex2png-ClassFieldTheory_247713527_-5.gif)
. We can further assume that
![$p\nmid \Char(K)$](./latex/latex2png-ClassFieldTheory_29624666_-5.gif)
. If
![$n=p^a$](./latex/latex2png-ClassFieldTheory_31754596_-4.gif)
and
![$y\in K_v$](./latex/latex2png-ClassFieldTheory_244718486_-4.gif)
such that
![$y^n=x$](./latex/latex2png-ClassFieldTheory_170030761_-4.gif)
. Then
![$K(y)/K$](./latex/latex2png-ClassFieldTheory_63070993_-5.gif)
is purely inseparable over
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
. But
![$K_v/K$](./latex/latex2png-ClassFieldTheory_20805283_-5.gif)
is separable for any
![$v$](./latex/latex2png-ClassFieldTheory_43021330_0.gif)
, which implies that
![$K(y)=K$](./latex/latex2png-ClassFieldTheory_48390929_-5.gif)
and
![$x\in (K^\times)^n$](./latex/latex2png-ClassFieldTheory_218136794_-5.gif)
.
First consider the case that
is cyclic of
-power order. We know that
splits into linear factors over
by the previous proposition. Now look at the factorization
over
and choose a root
of
in
for each
. If
for some
, then
for some
and
splits in
(notice that
is always abelian). Since
is a cyclic of
-power order, its subfields are totally ordered. Hence there exists an
such that
for any
. Therefore
splits in
for almost all
. By Corollary 10, we know that
and
has a linear factor
over
, i.e.,
.
The case
and
the follows since
is always cyclic of 2-power order, due to the assumption in the number field case and the fact that every finite extension of the constant field is cyclic extension in the global function field case. It remains to prove the case
odd and
. Notice that the extension
is always cyclic of
-power order and we can apply the first case to find
such that
. Let
, then
is an
-th power, where
is the degree of
. Because
is coprime to
, we know that
itself is an
-th power.
¡õ
Let us analyze the exceptional case in more detail. It is remarkable that we can write down the exceptional cases completely. In general when the local-global principle fails, it is quite rare that the failure can be completely classified.
Suppose
is not cyclic. We choose a
-th root of unity
such that
,
and
. Write
.
.
.
is cyclic of 2-power order.
Corresponding to the decomposition of
![$\Gal(\mathbb{Q}(\zeta_r)/\mathbb{Q})\cong \mathbb{Z} /2 \mathbb{Z}\times \mathbb{Z}/2 ^{r-2} \mathbb{Z}$](./latex/latex2png-ClassFieldTheory_198889147_-5.gif)
, we have the decomposition of
![$\mathbb{Q}(\zeta_r)=\mathbb{Q}(i)\cdot \mathbb{Q}(\eta_r)$](./latex/latex2png-ClassFieldTheory_23466932_-5.gif)
. The lemma follows immediately from this decomposition.
¡õ
Let
be the unique integer such that
and
.
![$K(\zeta_r)/K $](./latex/latex2png-ClassFieldTheory_87324124_-5.gif)
is cyclic for any
![$r\ge1$](./latex/latex2png-ClassFieldTheory_50558626_-3.gif)
if and only if
![$i\in K(\eta_{s+1})$](./latex/latex2png-ClassFieldTheory_252428509_-5.gif)
.
If
![$K(\zeta_r)/K $](./latex/latex2png-ClassFieldTheory_87324124_-5.gif)
is cyclic, then it contains a unique quadratic extension
![$K(\eta_{s+1})$](./latex/latex2png-ClassFieldTheory_110848797_-5.gif)
as
![$\eta_{s+1}^2=\eta_s+2$](./latex/latex2png-ClassFieldTheory_255587992_-6.gif)
. We find that
![$K(i)\subseteq K(\eta_{s+1})$](./latex/latex2png-ClassFieldTheory_58684641_-5.gif)
, hence
![$i\in K(\eta_{s+1})$](./latex/latex2png-ClassFieldTheory_252428509_-5.gif)
. If
![$i\in K(\eta_{s+1})$](./latex/latex2png-ClassFieldTheory_252428509_-5.gif)
, then
![$K(\eta_r)=K(\zeta_r)$](./latex/latex2png-ClassFieldTheory_61573527_-5.gif)
is cyclic for all
![$r\ge s+1$](./latex/latex2png-ClassFieldTheory_235697217_-3.gif)
by the previous lemma. For
![$r<s+1$](./latex/latex2png-ClassFieldTheory_233010848_-2.gif)
,
![$K(\zeta_r)$](./latex/latex2png-ClassFieldTheory_55997902_-5.gif)
is also cyclic because
![$K(\zeta_r)\subseteq K(\zeta_{r+1})$](./latex/latex2png-ClassFieldTheory_189416348_-5.gif)
.
¡õ
Let
![$S\subseteq V_K$](./latex/latex2png-ClassFieldTheory_47399490_-3.gif)
be a finite set. We define
![$P(m, S)=\{x\in K^\times: x\in (K_v^\times)^m, \forall v\not\in S\}$](./latex/latex2png-ClassFieldTheory_109002882_-5.gif)
.
The following key lemma characterizes all the counter examples.
![$K(\zeta_r)/K(i)$](./latex/latex2png-ClassFieldTheory_29540681_-5.gif)
is always cyclic, so by the Grunwald-Wang theorem there exists
![$y\in K(i)^\times$](./latex/latex2png-ClassFieldTheory_118151290_-5.gif)
such that
![$y^{2^t}=x$](./latex/latex2png-ClassFieldTheory_71718251_-4.gif)
. Let
![$\sigma\in \Gal(\mathbb{Q}(i)/\mathbb{Q})$](./latex/latex2png-ClassFieldTheory_56617072_-5.gif)
be the complex conjugation. We compute
![$$\sigma \zeta_s=\zeta_s^{-1},\quad \sigma \eta_s=\eta_s,\quad (y^{1-\sigma})^{2^t}=x^{1-\sigma}=1.$$](./latex/latex2png-ClassFieldTheory_237173365_.gif)
So
![$y^{1-\sigma}$](./latex/latex2png-ClassFieldTheory_206284080_-4.gif)
is a
![$2^t$](./latex/latex2png-ClassFieldTheory_131173356_0.gif)
-th root of unity. But
![$\bbmu_{2^t}\cap K(i)= \bbmu_{2^s}$](./latex/latex2png-ClassFieldTheory_106353866_-5.gif)
, hence
![$y^{1-\sigma}$](./latex/latex2png-ClassFieldTheory_206284080_-4.gif)
is a
![$2^s$](./latex/latex2png-ClassFieldTheory_131238892_0.gif)
-root of unity. So we can write
![$y^{1-\sigma}=\zeta_s^\mu$](./latex/latex2png-ClassFieldTheory_52436896_-4.gif)
for some
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
. Define an element
![$y_1=y\zeta_s^\lambda$](./latex/latex2png-ClassFieldTheory_7395814_-4.gif)
for some
![$\lambda$](./latex/latex2png-ClassFieldTheory_234755215_0.gif)
to be determined. Then
![$$y_1^{1-\sigma}=y^{1-\sigma}(\zeta_s^\lambda)^{1-\sigma}=\zeta_s^{\mu+2\lambda}.$$](./latex/latex2png-ClassFieldTheory_101057350_.gif)
If
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
is even, then we choose
![$\lambda$](./latex/latex2png-ClassFieldTheory_234755215_0.gif)
so that
![$y_1^{1-\sigma}=1$](./latex/latex2png-ClassFieldTheory_224254910_-4.gif)
. Then
![$y_1\in K^\times$](./latex/latex2png-ClassFieldTheory_22732983_-4.gif)
and
![$y_1^{2^t}=x$](./latex/latex2png-ClassFieldTheory_263607657_-4.gif)
(as
![$t>s$](./latex/latex2png-ClassFieldTheory_265456620_-1.gif)
), which contradicts our assumption. If
![$\mu$](./latex/latex2png-ClassFieldTheory_10449940_-4.gif)
is odd, we choose
![$\lambda$](./latex/latex2png-ClassFieldTheory_234755215_0.gif)
so that
![$y_1^{1-\sigma}=\zeta_s^{m'}$](./latex/latex2png-ClassFieldTheory_194892546_-4.gif)
. Replacing
![$y$](./latex/latex2png-ClassFieldTheory_43217938_-4.gif)
by
![$y_1$](./latex/latex2png-ClassFieldTheory_148671468_-4.gif)
, we may assume that
![$y^{1-\sigma}=\zeta_s^{m'}$](./latex/latex2png-ClassFieldTheory_3003141_-4.gif)
. Notice that
![$\zeta_s=\frac{1+\zeta_s}{1+\zeta_s^{-1}}$](./latex/latex2png-ClassFieldTheory_149993354_-11.gif)
, we compute
![$$y^{1-\sigma}=\zeta_s^{m'}=\left(\frac{1+\zeta_s}{1+\zeta_s^{-1}}\right)^{m'}=(1+\zeta_s)^{m'(1-\sigma)}.$$](./latex/latex2png-ClassFieldTheory_241356974_.gif)
If
![$z=y(1+\zeta_s)^{-m'}$](./latex/latex2png-ClassFieldTheory_101280633_-5.gif)
, then
![$z^{1-\sigma}=1$](./latex/latex2png-ClassFieldTheory_221549262_0.gif)
, hence
![$z\in K^\times$](./latex/latex2png-ClassFieldTheory_22902086_-1.gif)
. So
![$x=y^{2^t}=z^{2^t}(1+\zeta_s)^m=x_0z^{2^t}$](./latex/latex2png-ClassFieldTheory_4066745_-5.gif)
, where
![$$x_0=(1+\zeta_s)^m=(\zeta_{s+1}\eta_{s+1})^m=\eta_{s+1}^m.$$](./latex/latex2png-ClassFieldTheory_35902434_.gif)
Notice that
![$x_0=(\eta_s+2)^{2^{t-1}m'}\in (K^\times)^{m'}$](./latex/latex2png-ClassFieldTheory_229891949_-5.gif)
. Hence
![$x/x_0\in (K^\times)^{m'}$](./latex/latex2png-ClassFieldTheory_144831355_-5.gif)
. Also
![$x/x_0=z^{2^t}\in (K^\times)^{2^t}$](./latex/latex2png-ClassFieldTheory_118100536_-5.gif)
, hence the lemma follows.
¡õ
11/07/2012
We can actually show that
. If not, then
. Thus
for some
-th root of unity
. But
, which contradicts our assumption. Therefore if
then
is the disjoint union of
and
.
We have shown that in order to ensure
to be non-cyclic, we necessarily have
,
are non-squares in
. We shall show this obstruction always fails with the further condition (d) as in the following strong version of Grunwald-Wang theorem.
Suppose
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is a number field and
![$P(m,s)\ne (K^\times)^m$](./latex/latex2png-ClassFieldTheory_246960957_-5.gif)
, then (a)-(c) holds and
![$P(m,S)=(K^\times)^m\coprod x_0(K^\times)^m$](./latex/latex2png-ClassFieldTheory_55185784_-5.gif)
. We have to show that (d) holds, i.e., for
![$v\in S_0$](./latex/latex2png-ClassFieldTheory_112479126_-3.gif)
, then
![$x_0\not\in (K_v^\times)^m$](./latex/latex2png-ClassFieldTheory_134488703_-5.gif)
. If
![$x_0=\eta_{s+1}^m\in (K_v^\times)^m$](./latex/latex2png-ClassFieldTheory_149414430_-6.gif)
. Then we deduce that
![$(\eta_s+2)^{2^{t-1}}\in (K_v^\times)^{2^t}$](./latex/latex2png-ClassFieldTheory_263231750_-5.gif)
by Euclidean algorithm. Hence there exits
![$\zeta\in \bbmu_{2^{t-1}}$](./latex/latex2png-ClassFieldTheory_93295217_-4.gif)
and
![$\zeta(\eta_s+2)\in (K_v^\times)^2$](./latex/latex2png-ClassFieldTheory_182693297_-5.gif)
. If
![$-1\not\in (K_v^\times)^2$](./latex/latex2png-ClassFieldTheory_92406655_-5.gif)
, then
![$\bbmu_{2^{t-1}}\cap K_v^\times=\{\pm1\}$](./latex/latex2png-ClassFieldTheory_196939684_-5.gif)
, we know
![$\pm(\eta_s+2)\in (K_v^\times)^2$](./latex/latex2png-ClassFieldTheory_226651877_-5.gif)
. Otherwise,
![$-1\in(K_v^\times)^2$](./latex/latex2png-ClassFieldTheory_205139337_-5.gif)
, hence
![$v\not\in S_0$](./latex/latex2png-ClassFieldTheory_103902061_-4.gif)
.
For the other direction, we must show that if (a)-(d) hold, then
. It is equivalent to showing that
for any
. Let
. Condition (b) implies that
is a degree four extension with Galois group
. Moreover,
is unramified away from 2 by construction. If
is a palce of above
. Then
is quadratic. Consequently one of
is a square in
. If
and
, the one of
is a square in
by the assumption (d). In either case,
- if
or
is a square in
. Then
. Therefore
.
- if
, then
, hence
.
We conclude that
.
¡õ
11/30/2012
Outline of the proof
Group cohomology
Suppose
is any group. From an exact sequence of
-modules,
taking
-invariants gives an exact sequence
But the last map is not necessarily surjective, e.g., consider
and the Kummer sequence
The group cohomology functor
,
extends the above left exact sequence to the expected long exact sequence.
Dually, we can also take the
-coinvariants
(the largest quotient on which
acts trivially) which does not preserve injectivity and the group homology functor
fills in the corresponding long exact sequences.
When
is a finite group, Tate defines the norm map
. It maps into
and factors through
. Then we can connect both long exact sequence in cohomology and homology as
Write
,
and
. Then the five lemma gives a long exact sequence in Tate group cohomology
in both directions.
When
is cyclic, we further have
, i.e., the Tate cohomology groups are periodic of period 2.
Let
![$G=\Gal(L/K)$](./latex/latex2png-ClassFieldTheory_15982207_-5.gif)
be finite. Then the Hilbert 90 theorem asserts that
![$H^1(G,L^\times)=0$](./latex/latex2png-ClassFieldTheory_237925011_-4.gif)
. In particular, when
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
is cyclic,
![$\ker(N)/(\sigma-1)L^\times=\hat H_0(G,L^\times)=\hat H^{-1}(G, L^\times)=0$](./latex/latex2png-ClassFieldTheory_109428903_-4.gif)
, which implies that
![$N(x)=1$](./latex/latex2png-ClassFieldTheory_218340639_-4.gif)
if and only if
![$x=\sigma(y)/y$](./latex/latex2png-ClassFieldTheory_221818723_-4.gif)
(the classical formulation of Hilbert 90).
![$\hat H^0(G,L^\times)=K^\times/N(L^\times)$](./latex/latex2png-ClassFieldTheory_85718256_-4.gif)
.
![$\hat H^{-2}(G,\mathbb{Z})=H_1(G,\mathbb{Z})=G^\mathrm{ab}$](./latex/latex2png-ClassFieldTheory_222387151_-4.gif)
and dually
![$H^1(G,\mathbb{Z})=\Hom(G,\mathbb{Z})$](./latex/latex2png-ClassFieldTheory_76597989_-4.gif)
, where
![$\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_50353232_0.gif)
is endowed with the trivial
![$G$](./latex/latex2png-ClassFieldTheory_42562578_-1.gif)
-action.
Class formations
Class formations tell you all the group cohomology input in order to derive all the statements in class field theory (e.g., Artin maps).
Let
be a local or global field. Suppose
and
is a
-module. We say
is a continuous
-module if
(e.g.,
and
. Fix such an
, for any Galois extension
, we define
and
.
Suppose
is Galois and
, we have an inflation map
given by the precompostion by the natural surjection on cocycles.
Suppose
, we have a restriction map
given by the precompostion by the natural inclusion on cocycles.
Using pure group cohomology, we will prove the following main theorem.
Cup product with
![$u_{L/K}$](./latex/latex2png-ClassFieldTheory_200669351_-6.gif)
gives isomorphisms
![$\hat H^q(\Gal(L/K),\mathbb{Z})\cong \hat H^{q+2}(\Gal(L/K), A_L)=\hat H^{q+2}(L/K)$](./latex/latex2png-ClassFieldTheory_68373281_-4.gif)
. In particular, when
![$q=-2$](./latex/latex2png-ClassFieldTheory_268143558_-3.gif)
and
![$L/K$](./latex/latex2png-ClassFieldTheory_255685652_-5.gif)
is abelian, we obtain that
![$\Gal(L/K)\cong A_K/N(A_L)$](./latex/latex2png-ClassFieldTheory_18938555_-4.gif)
. We define the
Artin map ![$\Psi_{L/K}$](./latex/latex2png-ClassFieldTheory_31691942_-6.gif)
to be the inverse of this isomorphism.
Local class field theory
The main reference will be Serre's local fields and Galois cohomology. We set
, then
. The first axiom in class formation is simply Hilbert 90 and the second axiom amounts to proving that
. Let
(for local fields
, but this als0 works for general complete discretely valued fields with quasi-finite residue fields, however, the existence does not work, the local compactness is still needed). For any field
, we write
.
We have an exact sequence
![$$0\rightarrow \Br_k\rightarrow \Br_K\rightarrow\Hom_\mathrm{cont}(G_\mathrm{ur},\mathbb{Q}/\mathbb{Z})\rightarrow 0,$$](./latex/latex2png-ClassFieldTheory_82014675_.gif)
where
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
is the residue field of
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
.
The only non-formal part is the following
![$\Br_K=H^2(G_\mathrm{ur},(K^\mathrm{sep})^\times)$](./latex/latex2png-ClassFieldTheory_137751720_-4.gif)
(in other words, any central simple algebra over
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is split after an unramified base change.
When
![$k$](./latex/latex2png-ClassFieldTheory_42300434_0.gif)
is finite,
![$\Br_k=0$](./latex/latex2png-ClassFieldTheory_150662014_-2.gif)
(in other words, there are no noncommutative central division algebra over a finite field).
When
![$K$](./latex/latex2png-ClassFieldTheory_42824722_0.gif)
is local,
![$\Br_K\cong \Hom_\mathrm{cont}(\hat{\mathbb{Z}}, \mathbb{Q}/\mathbb{Z})\cong \mathbb{Q}/\mathbb{Z}$](./latex/latex2png-ClassFieldTheory_45859497_-4.gif)
.
The class formation machinery then proves local class field theory except the existence theorem. Proving the latter boils down to constructing enough abelian extensions using cyclotomic, Kummer and Artin-Schrier extensions (together with pure topological arguments).
Global class field theory
The main reference will be Cassels-Frolich and Artin-Tate. In global class field theory, the idele class group
plays the role of
in local class field theory. The first axiom in class formation is again Hilbert 90. For the second axiom, we will prove
is the the product of local Artin maps and it kills all the global elements. By the exact sequence
, the Brauer group of a global field
fits into
As we will see, the proof of the existence theorem (norm index inequality) interplays with the proof of Artin maps (class formation): first proved for cyclotomic extensions and then in general.