Math 791N. Algebraic Number Theory

Paul Gunnells, Farshid Hajir, Tom Weston, Siman Wong
Fall 2009. Tu/Th 2:30-3:45. Room 1322

Many classical problems in number theory concern rational solutions of polynomial equations with rational coefficients. But to solve these equations we need to introduce auxiliary algebraic numbers (i.e. roots of polynomials with Q-coefficients). In modern terminology this means working with finite field extensions of the rationals. Algebraic number theory is the study of properties of such fields. This course will cover the basics of algebraic number theory, with topics to be studied possibly including the following: number fields, rings of integers, factorization in Dedeking domains, class numbers and class groups, units in rings of integers, valuations and local fields, and zeta-and L-functions. Computational skills and examples will be emphasized, and there will be regular problem sets.

This course will be divided into four parts, each taught by one of the number theory faculty in the department. The instructor in charge of each part will determine the lecture format, text and homework requirement for that part. The following table summarizes the organization of the course; for additional information please consult the individual instructors.

We will make extensive use of the free computer algebra package PARI-GP. Here is an short tutorial and installation guide. PARI-GP will be used from the get-go, so get familiar with it as soon as possible!

Part I: Foundations

Date

Sept 8 - Sept 25 (tentative)

Instructor

Farshid Hajir

Topics

Number fields and rings of integers. Unique factorization. Ideal class groups. Units. Quadratic fields and quadratic forms.

Reference

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (2nd ed).

Additional Info

 
Part II: Frobenius and Ramification; Cyclotomic Fields

Date

Sept 29 - Oct 22 (tentative)

Instructor

Tom Weston

Topics

Galois action on primes. Frobenius. Inertia groups and decomposition groups. Cyclotomic fields.

Reference

Washington, Cyclotomic fields (tentative)

Additional Info

  
Part III: p-adic numbers; Local-Global Principles

Date

Oct 27 - Nov 12 (tentative)

Instructor

Siman Wong

Topics

Valuations, completions and p-adic numbers. Hensel's lemma. The Hasse principle; examples and counterexamples.

Reference

 Borevich-Shafarevich, Number Theory; Cohen, Number Theory I

Additional Info

 Problem Set #1 (Solution)     Problem Set #2
Part IV: Zeta Functions and L-functions

Date

Nov 17 - Dec 10 (tentative)

Instructor

Paul Gunnells

Topics

Zeta function of number fields and their residues. Dirichlet's theorem on primes in arithemtic progression. Factorization of zeta functions and prime decomposition. Class number formula (for quadratic fields).

Reference

 Neukirch, Algebraic Number Theory (tentative)

Additional Info