Time: Tuesday-Thursday, 10:10-11:25, room 307
Office Hours: Room 521, Tuesday 2-3, Wednesday 1-2 (Note change)
TA: Sam Mundy, s.mundy@columbia.edu
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:
-
Algebraic integers, factorization, Dedekind rings, local rings
(Gauss's first proof of quadratic reciprocity) -
Units and class groups (Pell's equation, classification of binary quadratic forms)
-
Cyclotomic fields (Fermat's last theorem for regular primes, first case;
Gauss's fourth proof of quadratic reciprocity) -
Congruences and p-adic numbers (the Chevalley-Warning theorem)
-
Zeta and L-functions (Dirichlet's theorem on primes in an arithmetic progression)
-
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.
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 now scheduled for October 31, during class. (NOTE CHANGE OF SCHEDULE)
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)
7th-week (due October 24)
(Midterm: no homework)8th-week (due November 7)
9th-week (due November 14 )
10th-week (due November 21)
11th-week (due December 5)
The final from the fall of 2013 is available here.
The final from the fall of 2015 is available here.
The final from the fall of 2017 is now available final2017.