MAT 1200/415 (Algebraic Number Theory); main reference: Milne's notes
Office hours: Thursdays 4-6pm (no office hours on January 30)
Homework 1 (due Feb 27): Marcus chapter 2, exercises 11, 28, 30, 40, 41, 42(ab).
Homework 2 (due Apr 4):
Determine the class groups of Q(root(d)) for d equal to 6, 10, -19, -23, -14, -21.
Show that the class number of Q(a) is 1, where a^3-a+1 = 0.
Marcus chapter 5, exercises 33, 47. Determine the unit group of O_K where K = Q(root(d)) for d equal to 5, 14.
Marcus chapter 3, exercises 16, 17, 30. Marcus chapter 4, exercises 5, 6, 7.
Homework 3 (due Apr 18):
Let K be the 527-th cyclotomic field. Determine all quadratic and all cubic subextensions of K.
Give a description of the primes that split completely in the cubic subextensions.
Cassels chapter 4, exercises 3, 5, 6. Also, answer the following question: for the cubic
field K = Q(a), where a has minimal polynomial F(X) given in Cassels exercise 6, find all e_i and f_i for p = 2, 3, 5, 7.
Adequate subgroups
(with Robert Guralnick,
Richard Taylor, and
Jack Thorne).
Appendix to On the automorphy of l-adic Galois representations with small residual image by Jack Thorne.
Journal de l'Institut de Mathématiques de Jussieu 11 (2012), no. 4, 907-920.