MATHEMATICS W4043, Fall 2013

Algebraic Number Theory

This is an introduction to the algebraic theory of numbers. The fundamental

techniques of the subject will be accompanied by the study of examples

of families of Diophantine equations that motivated the development of the subject.

Each of the topics listed below will occupy roughly two weeks of course time:

1. Algebraic integers, factorization, Dedekind rings, local rings

(Gauss's first proof of quadratic reciprocity)

2. Units and class groups (Pell's equation, classification of binary quadratic forms)

3. Cyclotomic fields (Fermat's last theorem for regular primes, first case;

Gauss's fourth proof of quadratic reciprocity)

4. Congruences and p-adic numbers (the Chevalley-Warning theorem)

5. Zeta and L-functions (Dirichlet's theorem on primes in an arithmetic progression)

6. Other topics (depending on class interest: Dirichlet's unit theorem, the prime number theorem,

cubic equations…)

Prerequisites: Basic algebra through Galois theory. Some elements of complex

analysis may be admitted in section 5.

Textbook: Marc Hindry, Arithmetics (Springer, 2011 edition)

Other useful references include

Dan Flath, Introduction to Number Theory

Pierre Samuel, Algebraic Theory of Numbers

Jean-Pierre Serre, A Course in Arithmetic

Midterm: October 22

Final: FINAL EXAM (online December 2)

Homework assignments

1st week (due September 12)

2nd week (due September 19)

3rd week (due September 26)

4th week (due October 3)

5th week (due October 10)

6th week (due October 17)

(Midterm: no homework)

7th week (due October 31)

8th week (due November 7 — two pages)

9th week (due November 14 — two pages)

10th week (due November 21)

11th week (due December 5)