Algebraic Number Theory

Professor A.J. de Jong, Columbia university, Department of Mathematics.

This semester I am teaching the undergraduate course on algebraic number theory. If you are interested, please email me and I will add you to the email list.

Lectures: Tuesday and Thursday 1:10 PM -- 2:25 PM in 307 math.

My office hours: Tuesday 9 - 10 AM in 523 math. Most Tuesdays I will be available also between 10 and 11 AM and you are welcome to come by during those times if my door is open and I am in my office.

TA: Morena Porzio, office hours / help room hours: 5 PM -- 8 PM on Wednesday in math 406.

Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent. This includes: Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups, rings, homomorphisms, ideals, integral and Euclidean domains, the division algorithm, principal ideal and unique factorization domains, fields, algebraic and transcendental extensions, splitting fields, finite fields, Galois theory.

Exams: There will be a take him midterm in the form of a longer set of exercises to take home in the midterm week (so due on Thursday, March 9). There will be a final exam.

Grading: grades will be computed using scores on weekly problem sets, midterm problem set, and final exam. The final exam will be worth 40% and the other 60% will be from the weekly problem sets with a higher weight for the one due on March 9 (which means it'll be worth roughly 12% and the other ones roughly 4% each).

Material: Online and offline texts to use:

Lectures: It is very important to be present during the lectures. Also, please keep up by reading the notes.

  1. Tue Jan 17: Section 1 of lecture notes. If you want another discussion of the material we talked about, then take a look at this writeup by Keith Conrad.
  2. Thu Jan 19: Section 2 and Section 3 up to and including Proposition 3.3 of the lecture notes. Note: we have changed the defn of a Euclidean domain. For more about Euclidean domains see this writeup of Keith Conrad (in particular the definition I gave in the lecture agrees with Definition 1.2 in this document).
  3. Tue Jan 24. Rest of Section 3 and material in Section 4 till end of proof of Thm 4.19 of the lecture notes. Extra materials: Cayley-Hamilton for rings, integral closure is a ring in the Stacks project, Cayley-Hamilton over a ring in the Stacks project.
  4. Thu Jan 26. Finish Section 4 and started with Section 5 up to and including the statement of Theorem 5.7 and the statement and proof of Lemma 5.8. Then we discussed Tr_{L/K} (trace) and Nm_{L/K} (norm) for finite extensions of fields, their computation in terms of sums and products of conjugates. We also stated and discussed the proof of the following Lemma: Let α be an element of a number field K. Then the following are equivalent: (1) α is an algebraic integer, and (2) the characteristic polynomial of multiplication by α : K → K as a Q-linear map has coefficients in the integers Z. End Lemma Some additional information you can read if you are interested:
  5. Tue Jan 31. Rest of Section 5 and Section 6 up to and inclusing Example 6.17. We also very carefully proved the statement of Exercise 3(ii) of hmwk2.pdf in the special case of Z[√ d] where d is a squarefree integer not 0, 1.
  6. Thu Feb 2. Finish Section 6 and Section 7 (entirely) and Section 8 up to and including 8.4.
  7. Tue Feb 7. Continuing with Section 8 of lecture notes up to and including Theorem 8.11.
  8. Thu Feb 9. Continuing with Section 8 and finished Section 9.
  9. Tue Feb 14. Section 10 and Section 11 until the proof of 11.1 including Corollary 11.5.
  10. Thu Feb 16. Section 11 and beginning of Section 12. Here are some links to material to read:
  11. Tue Feb 21. Discussed some of the details going into the proof (last time) of Theorems 11.6 and 11.9. Talked about some tricks with determinants of matrices. Here is a link pointed out by Jeff Kornhauser on determinants and matrices in block form. Then we discussed some material on modules over rings, homomorphisms between modules over rings, the abelian category of modules over a fixed ring, modules over fields are vector spaces, definition of Noetherian modules, Lemma 12.8, definition of Noetherian rings, Theorem 12.12. We gave a lot of examples of Noetherian rings, including: the integers, any polynomial algebra over a Noetherian ring, any quotient of a Noetherian ring, any field, the ring of integers of any number field, any subring of the ring of integers of a number field. Finally, we mentioned the existence of maximal ideals in nonzero rings and we proved Proposition 12.4.
  12. Thu Feb 23. Material on Dedekind rings, Section 13 and including 14 including proof of 14.1.
  13. Tue Feb 28. Rest of Section 14, Section 15, and Section 16.
  14. Thu Mar 2. Section 17. For Dedekind's factorization criterion see this handout Brian Conrad or see this expository note by Keith Conrad.
  15. Tue Mar 7. Section 18. Skip Section 19.1 -- 19.8. Localization: 19.9 ff.
  16. Thu Mar 9. Midterm problem set due! Localization continued: Section 20 and Section 21: Theorem 21.10 is an important generalization of Theorem 17.1.
  17. Tue Mar 14 -- spring break
  18. Thu Mar 16 -- spring break
  19. Tue Mar 21. Statement and proof of Theorem 21.10. Discussion of the example on the quadratic subfield of Q(zeta_p) explained at the start of Section 22. Discussion of Example 22.1.
  20. Thu Mar 23. Rest of section 22 on discriminant ideals. We will construct/define the discriminant ideal a bit differently: we let disc(A'/A) be the ideal of A generated by the expressions det(Tr_{F'/F}(e_ie_j)) where e_1, ..., e_n is any set of n = [F':F] elements of A'. Then we will show this gives the same thing as discussed in Section 22 of the lecture notes.
  21. Tue Mar 28. Section 23 (skip 23.1).
  22. Thu Mar 30. End Section 23 and Section 24.
  23. Tue Apr 4. Section 24 (starting on page 125; stuff about the class group of a number field) and then material on the geometry of numbers: Theorem 27.9 and Section 28 (not yet done completely).
  24. Thu Apr 6. Proof of Theorem 27.9 and discussion in the case of number fields: page 136 in the lecture notes, Lemma 26.3 and some stuff from Section 27.
  25. Tue Apr 11. Proof finiteness of class group. Sketch proof of lower bound of discriminants. See Sections 27 and 28.
  26. Thu Apr 13. Proof that there are finitely many number fields with bounded discriminant, see Theorem 28.4.
  27. Tue Apr 18. Proof of unit theorem, see Section 29.
  28. Thu Apr 20. Finding fundamental units in real quadratic fields. Quadratic reciprocity, discussion and proof. Both topics are briefly mentioned in Section 24.
  29. Tue Apr 25.
  30. Thu Apr 27.

Problem sets:

  1. First problem set due Thursday Jan 26 in class: do Exercises 1, 2, and 3(ii) and 3(iii) from hmwk1.pdf Please carefully read and comprehend 3(i) and use it in answering the other parts of 3.
  2. Second problem set due Thursday Feb 2 in class: do Exercises 1(all parts), 2(i), 2(ii), and 3(i) from hmwk2.pdf. Observe that 3(i) shows that an element of a number field is an algebraic integer if and only if its minimal polynomial over Q has integer coefficients. Here Q is the field of rational numbers. Please carefully read and comprehend 3(ii).
  3. Third problem set due Thursday Feb 9 in class: do Exercises 0(i), 0(ii), and 1 from hmwk3.pdf and try to do all of Exercise 2 of hmwk4.pdf.
  4. Fourth problem set due Thursday Feb 16 in class: do Excercises 3(i), 3(ii), 3(iii) from hmwk4.pdf and do Excercises 1(i), 1(ii), 1(iii) where phi is the Euler function (as in lecture) from hmwk5.pdf.
  5. Fifth problem set due Thursday Feb 23 in class: do Exercises 4(i), 4(iii) -- try to find your own argument without using the hint -- from hmwk5.pdf and all of Exercise 2 from hmwk6.pdf.
  6. Sixth problem set due Thursday, March 2 in class:
    1. Show that if A is a Dedekind domain and P is a nonzero prime ideal of A and x is an element of P not contained in P^2, then xA + P^2 = P. (Hint: Consider the prime factorization of xA + P^2 and eliminate the other primes.)
    2. Do Exercises 5(i), 5(ii), and 5(iii) from hmwk5.pdf
    3. Do exercise 3(i) from hmwk6.pdf.
  7. Midterm problem set due on Thursday, March 9: midterm problem set
  8. Seventh problem set due Thursday, Mar 23. Do exercises 1(i) and 1(ii) from hmwk7.pdf. Also do the following problems:
    1. Let f(x) = x^4 + a x^3 + b x^2 + c x + d with a, b, c, d integers. Let p be a prime number dividing c, d and assume p^2 does not divide d. Show that f does not factor completely into linear factors. (Modified Eisenstein criterion.)
    2. Let f(x) = x^d + a_1 x^{d - 1} + ... + a_d with integer coefficients. Assume that f is irreducible over Q and that f is Eisenstein at a prime number p. Let alpha be a complex root of f and K = Q(alpha). Show that p does not divide the index of Z[alpha] in the ring of integers O_K of K. Hints can be gotten by looking at exercise 3 of hmwk7.pdf linked above.
  9. Eigth problem set due Thursday, Mar 30.
    1. Let A be a discrete valuation ring with fraction field F. Let m = p be the maximal ideal of A. Assume the characteristic of the residue field A/m = A/p is not equal to 2 (you may use that this means that 2 maps to a unit in A). Let alpha be an element of A which has valuation either 0 or 1 and is not a square in F. Let F'/F be the degree 2 extension you get by adjoining a square root beta of alpha. Give a detailed proof that the integral closure A' of A in F' is equal to A' = A[beta].
    2. Do exercises 1(i) and 1(ii) of hmwk8.pdf
    3. Do exercises 4(i) and 4(ii) of hmwk8.pdf
  10. Ninth problem set due Thursday, Apr 6.
    1. Let K be the number field you get by adjoining the square root of 2 and of 3 to Q. Then K/Q only ramifies at 2 and 3 and Gal(K/Q) = Z/2 x Z/2 (up to you to make identification). Compute Frob_p for p = 5, 7, 11, 13, 17, 19, 23. Make a guess for Frob_p for all primes.
    2. Let f = x^3 - 4*x^2 + 3*x + 1. Let alpha be a complex root of f and denote K = Q(alpha).
      1. Show that f is irreducible over Q.
      2. Show that a prime p ramifies in K only if p = 7.
      3. Show that beta = (alpha - 3)/(alpha - 2) is a different root of f.
      4. Show that there is an automorphism sigma of K sending alpha to beta.
      5. Show that K/Q is Galois with Galois group G = {1, sigma, sigma^2}
      6. Compute Frob_p as an element of G for p = 2, 3, 5, 11, 13.
    3. Optional: Find a Galois extension K/Q which is cyclic of degree 4, produce an explicit isomorphism Gal(K/Q) = Z/4Z, and compute Frob_p for at least two unramified primes.
  11. Tenth problem set due Thursday, Apr 13.
    1. Do exercise 1(ii) in hmwk6.pdf
    2. Do exercises 1(i) and 1(ii) in hmwk9.pdf
    3. Do exercise 3 in hmwk9.pdf
    4. Optional: find an everywhere unramified degree 2 extension of Q(sqrt{15}).
  12. Eleventh problem set due Thursday, Apr 20. This may be a lot of work; I suggest just skipping an exercise if you are not interested in it!
    1. Let g > 1 be an integer. Let n > 1 be an odd integer. Assume that d = ng - 1 is squarefree. (For example g = 11, n = 3, and d = 177146.) Prove that the class group of Q(sqrt{-d}) has an element of order g.
    2. Compute the class group of Q(sqrt{14}).
    3. Let p be a prime number congruent to 1 mod 4 and congruent to -1 mod 3. Assume p > 3n for some n > 1. (For example p = 557 and n = 5.) Prove that the class group of Q(sqrt{-p}) has an element of order greater than n.
    4. Finiteness of class groups in the function field case: do the exercises in finite-class-function-field.pdf
  13. Twelth problem set due Thursday, Apr 27. Do exercises 18, 21, and 22 in Section 5 of these lecture notes by Peter Stevenhagen.