**Algebraic
Number Theory**

Room:

Time: 10:10-11:25, Monday and Wednesday

Instructor: Michael Harris

Office
Hours: Monday and Wednesday,
11:30-12:30 and by appointment, room number 521

Teaching
Assistant: Samuel Mundy

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
equationsâ€¦)

**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*

The
grade will be based on homework (20%), the midterm (30%), and the take-home
final (50%).* *

Midterm: October 25

Final: FINAL EXAM
(online December 11, due December 18)

** **Homework assignments**
**1st
week (due September
13)

2nd week (due September 20)

3rd week (due September 27)

4th week (due October 4)

5th week (due October 11)

6th week (due October 18)

(Midterm: no homework)

7th week (due November 1)

8th week (due November 8)

9th week (due November 15 )

10th week (due November 29)

11th week (due December 6)

The final from the fall of 2013 is available here.

The final from the fall of 2015 is available here.

**Return to Michael Harris's home
page**