Time: Tuesday-Thursday, 1:10-2:25, room 307 Mathematics

Office Hours: Room 521, Tuesday 2:30-3:30, Wednesday 1-2, and by appointment

TA: Alan Zhao, asz2115@columbia.edu, Help room hours Tuesday 6-8 PM

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, though not necessarily
in precisely that order:

1. Quadratic fields, algebraic integers, factorization, Dedekind rings, local rings

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 time and 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.

The course will roughly follow these course_notes.

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

Some of the homework exercises will be taken from a second book:
Daniel Marcus, Number Fields (Springer, 2018 edition, available at the Columbia Library website)

Other useful references include

Dan Flath, Introduction to Number Theory
Pierre Samuel, Algebraic Theory of Numbers
Jean-Pierre Serre, A Course in Arithmetic
Martin Weissman, An Illustrated Theory of Numbers
David A. Cox, Primes of the Form x^2+ny^2: Fermat, Class Field Theory, and Complex Multiplication

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

The midterm is tentatively scheduled for March 8, during class.

Homework assignments

1st-week (due January 27)

2nd-week (due February 3)

3rd-week (due February 10)

4th-week (due February 17)

5th-week (due February 24)

6th-week (due March 3)

``         (Midterm:  no homework)``

7th-week (due March 24)

8th-week (due March 31)

9th-week (due April 7 )

10th-week (due April 14)

11th-week (due April 28)

The final from the fall of 2013 is available here.

The final from the fall of 2015 is available here.

Here are the finals from the fall of 2017 final2017 and from
the fall of 2019 final2019.