**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)

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