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
, the primes
is equidistributed in
. Notice that there is a canonical isomorphism between
and
and this isomorphism sends the Frobenius element associated to any
to
. So the classical theorem of Dirichlet can be viewed as a special case of the following Chebotarev's density theorem.
Let
be a finite Galois extension. Then the Frobenius elements (conjugacy classes)
for primes
of
are equidistributed on the conjugate classes of
.
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
where
is a number field and
is a finite complex vector space. One can attach an
Artin -function to each Artin representation. When
is one-dimensional, an Artin representation is simply a character of
, which must factor through
. So a one-dimensional Artin representation is nothing but a character of
. By continuity, this character factors through
, where
is a finite abelian extension of
.
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
be a smooth projective connected curve over a finite field and
be the function field of
. Then class field theory classifies all abelian covers of
. In particular, this gives the abelianization of the etale fundamental group
of
. 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
. It is an abstract field and have many embeddings into the complex numbers
(we do not specify one).
A
global function field is a finite (separable) extension of
. It is a fact that the algebraic closure
of
in a global function field
is a finite field, called the
field of constants. Equivalently,
is the field of rational functions on a smooth projective geometrically connected curve
over
, unique up to isomorphism. The geometrically connectedness ensures that
is the field of constants.
A global field is either a number field or a global function field.
Let
be a global field. A
place of
is an equivalent class
of nontrivial absolute values
on
. Two absolute values
and
are equivalent if and only if they induce the same topology on
under the metric
, if and only if
for
. The set of all places of
is denoted by
. Suppose
, then we have the
completion of
with respect to any absolute values
corresponding to
.
When
, there is only one archimedean place
given by the usual absolute value and
. For any prime number
, the
-adic absolute value
is defined on the generators of
by
In this way we have a bijection between the set of primes of
and all non-archimedean places of
. The completion with respect to
is denoted by
.
Suppose
is an archimedean place of
, then there exists an absolute value
such that
for any
. By the Gelfand-Mazur theorem,
is either
or
, so the absolute value
is the usual absolute value on
or
. The
normalized absolute value is often defined as
(when
is real) and
(when
is complex). This normalization simplifies many statements like the product formula.
Suppose
is a non-archimedean place of
. Let
be a representative of
. Then
is discrete (Ostrowski's theorem), hence is equal to
for some
. We normalized the valuation
such that
. This valuation is intrinsic to
and is called the
normalized valuation. It extends uniquely to
.
We denote
, the
valuation ring of
;
, the
maximal ideal of
and
, the
residue field of
(a finite field). We define the
normalized -adic absolute value to be
. It is equivalent to absolute value we started with.
(Product formula)
For
,
for almost all places
and
Let
be a nonempty finite set of places of
containing the set
of archimedean places. The
ring of -integer is
. This is a Dedekind domain. For
, we denote
,
and
.
Every maximal ideal of
is of the form
for
.
For
,
. For any prime
,
.
Let
be a number field.
is called the
ring of integers of
. It is the integral closure of
in
.
Let
. Then taking the usual degree of rational function gives exactly the valuation associated to the closed point
. Moreover,
. A homomorphism
gives a map
. We know that
and
is the integral closure of
in
.
09/10/2012
Let
be a global field and
be a finite set of places of
containing
. Then
is a Dedekind domain. We denote the group of fractional ideals of
by
. Then
. Any
generates the principal fractional ideal
. Let
be the subgroup of principal ideals. We define the
-class group of
to be
.
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
. The number
is called the
ramification index of
, denoted by
. The number
is called the
residue degree of
. We have
.
is called
unramified at
if
for all
's,
totally split at
if
and
inert at
if
and
.
Now fix an archimedean place . Then is either 1 or 2.
is called
unramified if
and
ramified otherwise. Similarly we have
.
Suppose is Galois with , then acts on given by . This action is transitive on the fibers of .
The
decomposition group of
is defined to be the stabilizer of
in
. We have
. In particular, when
is abelian the
's for all
coincide, we simply denoted it by
.
When
is archimedean and
is ramified, we denote
, where
is the complex conjugation. These complex conjugations are related by
.
When
is non-archimedean,
acts on the residue extension
and gives a map
. This map is surjective and the kernel is called the
inertia subgroup, denoted by
. Similarly we have
. Counting shows that
. So
is unramified if and only if
, if and only if
.
Suppose
is unramified. The generator
of the cyclic group
is called the
Frobenius, denoted by
(you may think it as the analogue of the complex conjugation
. Similarly we have
. In particular, when
is abelian we have a unique Frobenius attached to
, denoted by
.
Valued fields
A
valued field is a pair
, where
is a field and
is a nontrivial absolute value on
. We endow
with the metric
. We say that
is
complete if
is complete under this metric.
A
valued field extension is a map
, i.e., a homomorphism
such that
.
(Gelfand-Mazur)
If
is a complete archimedean field. Then
is isomorphic to either
or
.
Let
be a non-archimedean valued field. The ring
is called the
ring of integers. It is a valuation ring with valuation group contained in
. Denote its fraction field by
. We similarly define the
maximal ideal and the
residue field .
We say
is
discretely valued if
is discrete (equivalently, equal to
for some
). Any element
such that
is called a
uniformizer. Let
be the unique valuation such that
.
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
is monic and
such that
and
. Then there exists
such that
and
.
09/12/2012
Let
be a complete discretely valued field and
be a complete discretely valued field extension of
. Let
be a uniformizer of
. We define
to be the
ramification index of
and
the
residue degree. We say that
is
unramified if
and
is separable (so
is separable), or equivalently,
is etale. We say that
is
totally ramified if
. Notice that
and
are multiplicative in towers
. So
is unramified (resp., totally ramified) if and only if
and
are unramified (resp. totally ramified).
We say
is
tamely ramified if
and
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
. So
and
are locally compact Hausdorff topological group.
Let
be a non-archimedean local field and
be a finite Galois extension with
.
acts on
by isometries, so
preserves
for any
and the action induces a map
. We define
to be the kernel of the map
, called
higher ramification groups (in the lower numbering). The higher ramification groups give a filtration
is called the
inertia group. We have
,
. In particular,
and
is totally ramified.
The fixed field of
is the maximal tamely ramified sub-extension
and
is the prime-to-
part of
.
is called the
tame inertia group and
is called the
wild inertia group.
is cyclic of order prime to
and
is the unique
-Sylow subgroup of
.
For
we define a function
, where
,
and
for
. Then
is a piecewise linear continuous increasing convex-up function. We define
to be the inverse function of
and define the
higher ramification groups (in the upper numbering)
. Then
and
, and
for
. For more details, see Chapter 4 of Serre's
local fields.
Topological groups
A continuous map
of topological space is called
proper if
is compact for any
compact,
open if
is open for any
open,
closed if
is closed for
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
, we endow the coset space
the
quotient topology, i.e.,
is open if and only if
is open. By definition, if
is continuous homomorphism, then
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
be a locally compact Hausdorff topological group. A
Haar measure on
is a nonzero Radon measure
on
such that
for any
and any measurable subset
of
.
(Haar)
The Haar measure exists and is unique up to scalar multiplication.
The Lebesgue measure on
satisfies
and is a Haar measure. The standard measure on
is a Haar measure.
Let
be any local field and
be a Haar measure on
. For
, set
, then
is also a Haar measure. It follows from Haar's theorem that
, where
is a constant.
.
The archimedean case is obvious. Suppose
is non-archimedean. Since
is compact open, we know that
is a positive real number. Thus it is enough to show that
, Replacing
by
, we may assume that
. It easy to see (via the filtration
) that
. Hence
is a disjoint union of
cosets of
. Therefore
using the left-invariance of
.
¡õ
Let
be local field and
be a Haar measure. Then
is a Haar measure on
, i.e.,
is left-invariant.
Profinite groups
A
profinite group is a (filtered) inverse limit of finite groups
. We endow
with the product topology, which makes it a compact and Hausdorff topological group. Then
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
are continuous). Moreover, if
then there exists a projection
, hence
and
are disjoint open and closed subsets, we conclude that
is
totally disconnected.
In fact, we can characterize the profinite groups topologically as follows.
A topological group
is profinite if and only if it is compact and totally disconnected. In this case,
, where
runs over all open normal subgroups of
Let
be any group. Then the
profinite completion of
is defined to be
, where
runs over all normal subgroup of finite index of
.
is profinite and the natural map
has dense image. Any homomorphism of
, for
profinite, factors through
.
Let
be a field, then
is a profinite group. Hence any homomorphism
factors as
.
.
For any non-archimedean local field
,
and
are profinite.
If
are profinite, then the product
is also profinite.
09/17/2012
Infinite Galois theory
Let
be a field. We say
is
separably closed if there is no finite separable extension of
. We define the
separable closure of
to be an algebraic field extension of
which is separably closed. Any two separable closure are isomorphic.
The
absolute Galois group of
is defined to be
.
(infinite Galois theory)
Suppose
is Galois. Then there is a bijection between subextension
and
closed subgroups of
under the Krull topology given by
and
. Moreover,
is Galois if and only if
is normal. In this case, we have
as topological groups.
We say
is
abelian if it is Galois with abelian Galois group
. In this case, it is easy to see that any subextension of
is abelian.
Let
be a topological group. Then
is a normal topological subgroup. The
topological abelianization of
is defined to be the Hausdorff topological group
, i.e., the maximal Hausdorff abelian quotient of
.
Now let be a global field and be a (possibly infinite) Galois extension.
For any
, there exists a place
of
such that
and
acts transitively on such
.
The finite case is known. Assume that
is infinite. We have a
-algebra homomorphism
induced by
. There is a unique absolute value
on
extending
and restricting
to
gives a
. Now suppose
and
are two such places of
. Then
is nonempty. So the inverse limit with respect to finite Galois extensions
is again nonempty.
¡õ
The
decomposition group at
of
is defined to be the stabilizer
. This is a closed subgroup of
.
Suppose
is non-archimedean. Let
be the the residue field of
representing
. This is an algebraic extension of
. Then
acts on
and give a continuous homomorphism
. This map at each finite level is surjective, hence the image is dense. But
is compact, so this map is actually surjective. The kernel
of this map is called the
inertia group. It is a closed subgroup of
and is equal to
. We say
is
unramified in
if
for any
. In this case, by surjectivity, we have a
Frobenius element whose image is
. When
is abelian, this doesn't depend on
and we have a Frobenius element
.
Define
. Then
, where
runs over all finite separable extensions of
. Moreover,
is Galois and the composition
is an isomorphism of topological groups, with all maps being bijective.
Let
, there exists
such that
. By Krasner's lemma, for
,
. Set
. Then
. Now since
, where
runs over finite Galois subextensions, we know that any
is Galois, hence
is Galois. The composition map
is an inverse limit of isomorphisms, hence is a topological isomorphism. The injectivity of
follows from the fact that
is dense in
.
¡õ
09/19/2012
Suppose
. Then
.
By definition,
. Let
be a monic irreducible separable polynomial and
be a root of
. Let
with coefficients close to
and the same degree as
. Then
is small. On the other hand, write
, where
. Then
can be made as small as possible. Now Krasner's lemma implies that
. Comparison of degrees shows that
. So
is separable and irreducible over
, hence over
. We conclude that
, thus
.
¡õ
.
is always open.
is proper, hence is closed. It is enough it show that
is open is
by Exercise
3. Since we already know that the image is closed, it suffices to show that
is of finite index. This follows from the norm-index formula. In fact, homological methods can show that
(and equality holds if and only
is abelian).
¡õ
The unramified case is relatively simpler.
If
is unramified, then
. Hence
.
is clear. Since the image is closed in
, it suffices to show that the image is dense. Any
acts by isometry on
,
. So
, where the last equality holds because
is unramified. The norm map induces a map
, 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
, which coincides with the trace map, so it is nonzero and
-linear (trace is always surjective for finite separable extension). This concludes that the norm map has dense image.
¡õ
Adeles
Let
be a family of topological spaces,
be open subset defined for almost all
. We define
and endow
with the topology given by the base of open subsets
The topological space
is called the
restricted topological product of
with respect to
. Notice that this topology is different from the subspace topology induced from the product topology.
The following lemma is easy to check.
Let
be a finite set of indices containing all
's such that
is not defined. Then
is open in
and the subspace topology on
is the product topology on
.
The following proposition partially explains the reason of introducing the restricted product.
If
's are locally compact Hausdorff and the
's are compact, then
is locally compact Hausdorff.
Notice that
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
.
¡õ
Let
be a global field. The
ring of ideles is defined to be the restricted product of
with respect to
for
non-archimedean. It is a subring of
and is a locally compact Hausdorff topological ring.
Since
, we have
. The similar identification works for general number fields when replacing
by
. We also have
.
is a discrete closed subgroup and the quotient
is compact.
09/21/2012
We will omit the tedious measure-theoretic check of the following the lemma.
There exists a Haar measure
on
such that
where
is the normalized Haar measure on
such that
.
Ideles
The group of
ideles is defined to be
. The embeddings
defined by
gives an embedding
. The quotient
is the called the
idele class group of
.
The
idelic norm is defined to be the homomorphism
given by
. The
unit ideles is defined to be the subgroup
.
- 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
be a global function field with constant field
and
. Then
.
For
, then multiplication by
scales
by
.
It follows from the case of local fields (Lemma
1) and the Lemma
6.
¡õ
is a Haar measure on
.
From this we can obtain a slick proof of the product formula.
09/24/2012
(Theorem 2)
Consider the measure
in Remark
42. Suppose
, then
Since
is
invariant under multiplication by
, this integral is equal to
Hence
.
¡õ
Let
. Then the fractional ideal
where
determined up to sign. Hence
. But
implies that
. Replacing
by
if neccesary, there exists a unique
such that
as well. Hence
is a fundamental domain for
. Therefore
as topological groups. Hence
as topological groups. Now
also embeds into
, hence
as topological groups (this can also be seen directly by extracting all
-powers of
).
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
be a
-dimensional
-vector space and
be a lattice. Suppose
is a Haar measure on
constructed from the counting measure on
and the volume one measure on
. If
is compact convex and symmetric around 0, then
implies that
.
The idea of the proof of the following adelic version is essentially the same as the classical version.
(Adelic Minkowski)
Let
be a global field and
. Then
is compact. There exists
depending only on
such that if
, then
.
(Strong approximation)
Let
and
. Then the diagonal embedding
has dense image.
We claim that there exists
such that
. This follows from the fact that
is open (Exercise
3),
and
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
is closed in
and the subspace topology on
from
coincides with the subspace topology from
.
First we show that
is closed in
. For
, the products
will eventually be decreasing. So
is well-defined. Suppose
, then
. 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
be a global field. Then
is a discrete closed subgroup and
is compact.
The assertion that
is discrete and closed follows form the case of
(Theorem
12) since the topology on
is finer than
. It remains to prove the compactness of
. By the previous lemma, if
is compact, then
is compact in
. So it suffices to show that there is a surjection
for some
compact. Let
be as in the Adelic Minkowski's Theorem
15 and choose
such that
. We claim that the compact set
works. Let
, then
. It follows from the Adelic Minkowski's Theorem
15 that there exists
, i.e.,
for any
. Now
, hence
. This concludes the surjectivity of
.
¡õ
Classical finiteness theorems
Let
be a finite set of places. We denote
an open subring of
. Similarly, we denote
an open subgroup of
. We have
and
.
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
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
is a closed embedding.
For any
,
. Hence
. When
is a global function field, we have two exact sequences
Since
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:
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 .
is continuous, open and proper.
It is continuous since each local norm is continuous and
(hence the inverse image of a basic open subset is open). For the properness, we use the splitting
and
. Then
is the norm on the compact factors
and identity on
or
, 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
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
is a number field. Then
is the connected component of 1 in
and
is profinite. Moreover,
is the set of all divisible elements in
.
Notice that
is finite and
, we know that the natural map
has finite cokernel. Let
be the image of this last map. Since
is profinite, we know that the image
is also a profinite group (Remark
30). Since
is of finite index, it is also open in
. Combining the fact that
is compact and totally disconnected, we find that
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
is a global function field. Then
is totally disconnected and has no nontrivial divisible elements.
Then
is an open neighborhood of 1 in
, hence
is an open neighborhood of 1 in
. Hence
is totally disconnected, thus
is profinite. Hence
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
For
,
and
. So
forms a filtered directed system. We define the
maximal cyclotomic extension to be
.
10/01/2012
When
or
, the Kronecker-Weber theorem says that
. However, this is not true for general number fields. For example, for
, the extension
is abelian over
but is not even Galois over
, hence
cannot be contained in a cyclotomic extension of
since
is abelian over
for any
.
Suppose
, where
. Suppose
and
is the order of
. Since
, we know that
. Therefore
and
is a degree
extension of
. We have
,
and
is given by
.
Suppose
and
, then
and
sends the complex conjugation to
as
.
Suppose
is a non-archimedean local field. If
, then
is unramified. Indeed, if
is any finite extension, Hensel's lemma implies that
if and only if
. Hence
is the unramified extension with residue field
. We have
as the case of finite fields.
Suppose
is a global function field, then
. However, it is not clear whether this is the maximal abelian extension (indeed, not in general).
Let
be a number field and
. Then
is ramified at most over finite places
and ramified at the real places if
. For
,
, where
, due to the compatibility of
with respect to the inclusion
, and the decomposition group
maps to
. For
,
maps to
for
. The question remaining is what
takes
for
(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
be a finite subextension. Then there exists a unique smallest
(the notation comes from German word
Führer) such that
. Moreover,
if and only if
is ramified in
.
Since
is finite, there exists an
such that
. Since
, the gcd of all such
's is the smallest
. If
, then
is unramified in
. Conversely, if
is unramified in
, we write
, then the restriction map on inertia groups is
. Hence
is contained in the fixed field of
of
, i.e.,
.
¡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
be a finite abelian extension
. Define the
Artin map by sending
to
and
as the restriction
. It is a continuous surjection. It follows that
maps surjectively to the inertia group
and
maps to
.
10/03/2012
,
. When
is unramified is
,
(opposite to the usual Artin map), where
is the image of
.
We already know that
. Without loss of generality we may assume
. Suppose
is unramified, then
maps to
, i.e.,
. For arbitrary
, we write
. Then
. Because any prime
is unramified in
, we also know that
. Hence
We know that
This completes the proof.
¡õ
Now for any
with infinite degree,
makes sense by taking the inverse limit over
finite over
.
is obviously continuous by construction. Since
is compact and the image is dense (surjective on every finite
, we find the
Artin map 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
We define the
Artin map sending
This is a continuous map with dense image and maps
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
such that
is open in
,
has the subspace topology under
and any splitting of
induces an isomorphism
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
be the abelianization of
and
:
Then
with
the abelianization as topological groups (but not for
).
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
and
gives a bijection between open subgroups of
and open finite index subgroup of
. For any
an open subgroup,
is isomorphic to
as discrete coset spaces.
This bijection extends to a bijection between closed subgroups
of
such that
or
has finite index in
, and closed subgroups of
. 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
.
Observe that if
is a finite set of places of
. Then
is dense in
by the weak approximation (Remark
47). If
is a finite abelian extension, we let
contain
and all the ramified places. Then
is determined by (a), hence by continuity
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)
is an inclusion reverse bijection between finite abelian extensions
and open subgroups of finite index in
.
It suffices to show the injectivity. Suppose
and
are finite abelian extensions such that
. Let
and
be the corresponding open subgroup in
, then
. Hence
. Suppose
, then there is an neighborhood
since
and
are both open and closed. But
is dense, this would contradict
. We conclude that
, thus
.
¡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
is a number field, then
induces an isomorphism of topological groups
.
We thus deduce a stronger version of existence theorem for number fields.
The map
is a bijection between
all abelian extensions
and
closed subgroups of
such that
. Under this bijection,
is called the
class field of
.
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
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
induces vertical isomorphisms of topological groups
So
can viewed as the Weil group
and
is the profinite completion of
.
gives a bijection between
all abelian extensions
such that the constant field extension
is either finite or equal to
, and
closed subgroups of
.
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:
(Norm functoriality)
The follows diagram commutes:
Suppose
is finite abelian, then
. In other words,
.
Take
in the previous theorem.
¡õ
(Global norm index inequality)
Suppose
is a global field and
is a finite and separable extension. Then
.
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
is a global field and
is a finite and separable extension. Then
if and only if
is abelian, in which case
.
It remains to prove the "only if" direction, which is left as an exercise (hint: the left-hand-side is always
.
¡õ
Now let us briefly turn to the verlagerung functoriality of the Artin map.
Let
be a group and
be a subgroup of finite index. For any
, let
be the smallest
such that
. This only depends the choice of
in
, a finite double coset. Write
be the coset representatives. We define the
verlagerung (or
transfer)
. Then
is a group homomorphism. In terms of group cohomology, this is the restriction map
and functorial in
.
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
is a global field and
is a finite separable extension. The following diagram commutes:
Suppose
is finite Galois and
. 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
is a local field and
is a finite and separable extension. Then
.
(Existence Theorem)
Suppose
is a local field and
is a finite and separable extension. Then
if and only if
is abelian, in which case,
.
The local-global compatibility will follow from defining the global Artin map via "gluing" local Artin maps.
(Local-global compatibility)
Suppose
is a global field and
. Then we have a commutative diagram
If
is finite abelian and
, then
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
is a number field and
is a real place. If
is finite abelian, then
is unramified in
if and only if
kills all
. Otherwise
kills
and
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
be a global field. A
modulus of
is a formal product
, where
and
for almost all
; for
a real place,
;
a complex place,
. We denote by
the
-component of
. We write
or
if
. The modulus is used to keep track of the ramification of the places of
in some sense.
Let
, we say
if and only if
for any
. This is a multiplicative condition, so it makes sense to say that
whenever
.
For
, each modulus is of the form
or
, where
. It follows easily that an idele
if and only the finite part
(which can be thought of the usual congruence relation
), and
if we further have
.
Suppose
is a modulus, we define
, namely
It is an open subgroup of
and
if and only if
. We know that
forms a cofinal filtered system of open subgroups in
. This generalizes Proposition
4.
For
and
, we have
. We have a surjection
and the kernel is exactly
. Therefore
. Similarly, when
, we find that
.
10/17/2012
There are isomorphisms
We define
(it properly contains
) and
. Then we have an injection
It is actually an isomorphism by weak approximation. Now define the homomorphism
which is obviously a surjection with kernel
. Taking quotient by the image of
, we obtain that
.
¡õ
The
ray class field of the modulus
is the class field
corresponding to
. When
is a number field,
is finite abelian. When
is a global function field,
contains
since the image of
has trivial image when projected to
.
When
, we have
and
and the following diagram commutes
We summarize the easy properties of ray class fields as follows.
- is unramified at all .
- If , then .
Suppose
is a finite abelian extension. We say a modulus
of
is
admissible for
if
. The gcd of all admissible modulus of
is called the
conductor of
. It follows that
.
When
.
is essentially the same as Proposition
3, except that
if and only if
is totally real.
is admissible for
if and only if
.
The "if" direction follows from the fact that
. For the "only if" direction, suppose
, then the local-global compatibility implies that
for
nonarchimedean (and a similar thing for archimedean places), hence by the definition of idelic norm,
. Therefore
.
¡õ
is ramified in
if and only if
.
Let
, where
. If
is ramified, the
, hence
, hence
, which implies that
. Now suppose
is unramified, then for any
,
. Then we can find a modulus
such that
and
. It follows that
by the previous lemma and the definition of
.
¡õ
Ideal-theoretic formulation of global class field theory
Let
be a finite abelian extension and
be a modulus which is divisible by all ramified places. We define the
Artin symbol
is admissible if and only if
. (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
is a finite extension and
is a modulus of
. We denote by
the free abelian group generated by prime ideals of
whose restriction to
are coprime to
. The usual ideal norm restricts to a group homomorphism
. We denote the image of
by
.
Let
be a finite extension. Suppose
is finite abelian and
. Then we have a commutative diagram
where
is a modulus of
divisible by the primes of ramified in
,
is a modulus of
divisible by primes ramified in
or restrict to primes of
ramified in
.
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 where
is the absolute norm.
We omit the proof of the simple analytic property.
is analytic on
except a simple pole at
.
Let
be a modulus of
and
be an ideal class. We define
A similar analytic property holds for too.
is analytic on
except a simple pole at
. The residue at
depends on the modulus
but not on ideal class
.
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
- If , then
The
Weber -function for a character
is defined to be
It follows from the previous theorem that
is analytic on
except a possible simple pole at
. When
, it is actually analytic at
by the previous proposition, since all the residues of
at
are the same.
Consider
and
. Then
is simply the
Riemann zeta function. A character
is the same thing as a
Dirichlet character . Then
is simply a
Dirichlet -function, where we extend
on
by letting
whenever
.
Now class field theory easily imply the following result on special values of Weber -functions.
if
.
By global class field theory, there exists a class field
such that
. So
can be viewed as a character of
. Let
be a modulus of
divisible exactly by the primes of
restricting to
. Then as a special case of the lemma below (
)
The result then follows since each of the
-functions
and
has a simple pole at
.
¡õ
10/22/2012
Let
be a finite abelian extension. Suppose
is an admissible modulus for
and
is exactly divisible by primes of
restricting to primes dividing
. Then
where
runs over all characters
.
This equality actually holds at the level of local Euler factors. Say
, then
is unramified and
, we claim that
Write
, then
. The right hand side becomes
Write
for short. Taking the logarithms of both sides, it reduces to show that
Notice that
and
has order
in
. Then the character
of the Pontryagin dual of
is nontrivial if and only if
. Therefore the right-hand-side is simply
which coincides with the left-hand-side.
¡õ
Consider a one-dimensional Galois representation
. Then
factors through a finite cyclic extension
, where
. Let
be the conductor of
, then
induces a character
. By definition,
since
if and only if
is unramified in
. In other words, Weber
-functions can be viewed as the same thing as one-dimensional Artin
-functions via
class field theory. Notice Weber
-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
-functions give handy ways to establish non-vanishing results of special values. This picture motivates the Langlands program of the study of general Artin
-functions via relating Galois representations and automorphic representations and class field theory can be viewed as the Langlands program for
.
Suppose
is a finite abelian extension with
. Then
(notice the left-hand-side is the regular representation of
). Thus we have
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
is a finite group and
. Then there exists
,
subgroups of
and characters
such that
as virtual representations.
With the same notation,
Chebotarev's density theorem
Let
be a number field of degree
. Let
be a set of finite primes of
. We define the
natural density of
to be the limit (if it exists)
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
is a number field. Denote the
degree of a prime
by
. Prove that
has density 1. (Not to be confused with the density of of split primes of
in
, which will be shown to be
).
Let
be a modulus and
. Then
.
Notice that
On the other hand, since
is analytic at 1 whenever
, we know that
Expanding the last sum gives
The desired result then follows.
¡õ
When
,
, we have
and an ideal class is given by simply given by a residue class modulo
. From the previous proposition we recover the classical theorem of Dirichlet on primes in progressions:
. Namely, the primes are equidistributed modulo
.
(Chebotarev's density)
Suppose
is a Galois (but not necessarily abelian) extension of global fields with
and
. Let
and
be the conjugacy class of
in
(with
). Then
has density
. In other words, the Frobenius conjugacy classes are equidistributed in
.
Let us first show the case that
is abelian. Let
be the conductor of
. Let
be the kernel of
. Then
and
, where
is any preimage of
. Since
contains exactly
ideal classes, we know that
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
be a global field and
be a finite separable extension. We define
For
, we also define
.
Suppose
is
Galois and
. Then
and
have density
.
Use Chebotarev's Density Theorem
34 for
.
¡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
has density 1, then
.
Apply the previous exercise to the Galois closure of
.
¡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
are finite
Galois,
. Then the following are equivalent:
- .
- .
- .
(a) implies (c) and (c) implies (b) are obvious. For (b) implies (a), notice that
. Therefore
and
.
¡õ
Suppose
and
are finite Galois,
. Then
if and only if
. In other words, a Galois extension is determined by the set of split primes.
Hilbert class fields
Let
be a number field. The
Hilbert class field of
is defined to be the ray class field
for
. Then
and
is unramified at every place (including archimedean places). The
narrow Hilbert class field of
is defined to be the ray class field
, where
is the product of all real places of
. Then
(the
narrow class group) and
is unramified at all finite places. Notice that
and
surjectis onto
.
(resp.
) is the maximal abelian extension of
which is unramified everywhere (resp. at all finite places).
If
is unramified everywhere, then
by Proposition
12. Hence
. Similarly for
.
¡õ
10/31/2012
Suppose
is finite extension of number fields and is totally ramified at some
. Then
.
Since
is abelian and unramified everywhere, we know that
. So it suffices to show that
and
are linearly disjoint over
(i.e.
. Suppose
with
the minimal polynomial of
. Let
be a monic polynomial dividing
over
. Since
is Galois,
splits into linear factors over
, which implies
also splits over
. Hence
, which is equal to
by the assumption that
is totally ramified at some place. Hence
and
is irreducible over
as needed.
¡õ
Consider
,
. We know that
using the previous proposition. When
is a prime, the famous criterion of Kummer asserts that
divides
(indeed equivalent to that
divides
) if and only if
divides the numerator of some Bernoulli number
. Such a prime
is called
irregular.
Artin's principal ideal theorem
(Principal Ideal Theorem)
Let
be a number field and
be its Hilbert class field. Then any ideal
becomes principal in
.
Using class field theory, it will reduce to the following purely group-theoretic theorem (we omit the proof).
(Furtwangler)
Let
be a finite group and
. Then
is trivial.
(Proof of the Principal Ideal Theorem)
Let
be the Hilbert class field of
. Then
is Galois since
is intrinsic to
. Notice that
is the maximal abelian subextension of
. Therefore
and
, where
. We have the following commutative diagram
By the previous theorem, we know that the
, hence
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
be a number field of degree
. If
, then
where
is the
-rank of a finite group of
.
On the other hand, Brumer's theorem provides a lower bound on .
(Brumer)
Suppose
is Galois of degree
. Let
be the number of primes
such that
for any
above
. Then
Suppose
is Galois of degree
. If
, then
.
When
, 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.,
where
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
be a Hausdorff group. A continuous character
is called
unramified if the class field corresponding to
is unramified at
.
Suppose
is a global function field of characteristic
and
is a finite extension of
for some
. Let
be a continuous homomorphism unramified outside a finite set
. Then there exists
and
continuous of finite order such that
.
Since
is abelian,
factors through
. Pick
such that
. Let
and
. Notice that
by construction. But
, it suffices to show that
is finite as
is finite. But
and by assumption
for
, it suffices to show that
is finite. But
has a finite index pro-
group and
has a finite index pro-
group. Because there is no nontrivial map from a pro-
group to a pro-
group, the image must be finite.
¡õ
Grunwald-Wang theorem
Let
be a global field. Suppose
and
contains the
-th roots of unity. If
is an
-th power in
for almost all
, then
itself is an
-th power in
.
Let
be an
-th root of
and
be its minimal polynomial. So
. If
is an
-th power in
, then
splits completely in
as
contains the
-th roots of unity. Since
is separable, it follows that
splits completely in
. Now the split primes
has density 1, hence
.
¡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
be a global field. If
is a
-th power in
for almost all
, then
is an
-th power in
, except potentially if
is a number field and
is not cyclic, where
with
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
is a power of a prime. In fact, suppose
with
coprime, then by the Euclidean algorithm we can write
. Suppose
, then
. We can further assume that
. If
and
such that
. Then
is purely inseparable over
. But
is separable for any
, which implies that
and
.
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
, we have the decomposition of
. The lemma follows immediately from this decomposition.
¡õ
Let be the unique integer such that and .
is cyclic for any
if and only if
.
If
is cyclic, then it contains a unique quadratic extension
as
. We find that
, hence
. If
, then
is cyclic for all
by the previous lemma. For
,
is also cyclic because
.
¡õ
Let
be a finite set. We define
.
The following key lemma characterizes all the counter examples.
is always cyclic, so by the Grunwald-Wang theorem there exists
such that
. Let
be the complex conjugation. We compute
So
is a
-th root of unity. But
, hence
is a
-root of unity. So we can write
for some
. Define an element
for some
to be determined. Then
If
is even, then we choose
so that
. Then
and
(as
), which contradicts our assumption. If
is odd, we choose
so that
. Replacing
by
, we may assume that
. Notice that
, we compute
If
, then
, hence
. So
, where
Notice that
. Hence
. Also
, 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
is a number field and
, then (a)-(c) holds and
. We have to show that (d) holds, i.e., for
, then
. If
. Then we deduce that
by Euclidean algorithm. Hence there exits
and
. If
, then
, we know
. Otherwise,
, hence
.
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
be finite. Then the Hilbert 90 theorem asserts that
. In particular, when
is cyclic,
, which implies that
if and only if
(the classical formulation of Hilbert 90).
.
and dually
, where
is endowed with the trivial
-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
gives isomorphisms
. In particular, when
and
is abelian, we obtain that
. We define the
Artin map 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
where
is the residue field of
.
The only non-formal part is the following
(in other words, any central simple algebra over
is split after an unramified base change.
When
is finite,
(in other words, there are no noncommutative central division algebra over a finite field).
When
is local,
.
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.