Math 676 Syllabus

# Math 676

Math 676 is a first semester graduate course in algebraic number theory.

## Course details

Time: MWF, 11AM-noon

Place: 4096 East Hall

First meeting: Wednesday, September 3, 2008

Jeffrey Lagarias
Email: lagarias@umich.edu
Web: http://www.math.lsa.umich.edu/~lagarias
Office: 3086 East Hall
Office Phone: 763-1186
Office Hours: TBA

## Course description

The fundamental theorem of arithmetic states that any non-zero integer is expressible as a product of primes in a manner that is unique up to order of the factors and units (i.e. signs). Algebraic number theory begins with trying to understand how this generalizes (or fails to do so) in algebraic number fields. An algebraic number field is a finite extension of Q, and an element of such a field is called an algebraic number. The course covers the basic structure of such field and some analogs in positive characteristic. In particular we will study algebraic integers (these play a role analogous to that of Z inside of Q) and we will prove two of the important finiteness theorems: finiteness of class grous (these measure the failure of the fundamental theorem of arithmetic for rings of algebraic integers) and finite-generatedness of unit groups (these generalize classical results centered on Pell's equation). We will include as many examples and applications as possible and some of these will be developed in the homework. The main topics to be covered are: (1) Valuations and local fields (2) Dedekind domains, global fields, rings of integers (3) Class groups (4) Unit groups (5) L-functions and class number formulas If time permits we may discuss the adele ring and idele group attached to a number field. The course (or at least a solid command of the material in it, ignoring characteristic p if you are analytically inclined) is absolutely essential for anyone wishing to study number theory.

## Textbooks

The basic textbook the course will use is:

• J. W. S. Cassels, Local Fields, Cambridge University Press 1986.

Students should get a copy of this textbook. It is available in paperback.

## Homework Assignments:

Homeworks are due approximately biweekly.
Students are expected to in solutions to 5 problems out of total;
doing more problems is optional.

Homework 1 (due Wednesday, Sept. 17 )

Homework 2 (due Wednesday, Oct. 1 )

Homework 3 (due Friday Oct. 17 )

Homework 3 (11th problem) (due Friday, Oct. 17 )

Homework 4 (due Wednesday, Nov. 5 )

Homework 4 (11th problem-mandatory) (due Wednesday, Nov. 5 )

Homework 5 (due Wednesday, Nov. 19 )

Homework 6 (due Wednesday, Dec. 3 )

Homework 7 (due Monday, Dec. 8 )

## Syllabus and Homeworks

Homeworks are approximately biweekly. This syllabus will be updated as the course progresses.

• ## Other Books

Other sources:

• J. Milne, Algebraic Number Theory, version 3.00 notes.

These notes assume knowledge of commutative algebra.

A good, inexpensive, book to get is:

• P. Samuel, Algebraic Theory of Numbers (Hermann, Paris 1970) Dover reprint 2008.

Another good book:

D. Marcus, Number Fields , Springer_Verlag, New York 1977

This book has a great selection of exercises.

Other good references:

• S. Lang, Algebraic Number Theory (Second Edition) Springer-Verlag: New York 1994

• G.Janusz, Algebraic Number Fields, 2nd Edition (AMS 2005)

A very good modern source is:

• J. Neukirch, Algebraic Number Theory (Springer-Verlag 1999-expensive).

For extensions to class field theory:

• Cassels and Frohlich, Algebraic Number Theory (excellent, but out of print)
• Serre, Local Fields (for local class field theory)
• Washington, Introduction to Cyclotomic Fields (for Kronecker-Weber, and some applications)

## Prerequisites

Graduate coursework in algebra is also recommended.
If you are unsure how your background matches these prerequisites, see me.

## Homework/Exams

TBA

Jeff Lagarias (lagarias(at)umich.edu)