**Algebraic
Number Theory**

Room:

Time: 2:40-3:55, Tuesday and Thursday

Instructor: Michael Harris

Office
Hours: Tuesday and Thursday, 10-11
and by appointment, room number 521

Teaching
Assistant: TBA

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 29

Final: FINAL EXAM
(online December 14, due December 21)

** **Homework assignments**
**1st
week (due September
17)

2nd week (due September 24)

3rd week (due October 1)

4th week (due October 8)

5th week (due October 15)

6th week (due October 22)

(Midterm: no homework)

7th week (due November 5)

8th week (due November 12)

9th week (due November 19 )

10th week (due December 3)

11th week (due December 10)

The final from the fall of 2013 is available here.

Course notes (will be frequently updated)

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