# Math 125 (Spring 2011): Number Theory

Quick links: News | Introduction | Homework | Instructor information | Grading | Textbook

## News

• 3/28/2011: This webpage goes up!

## Introduction

This semester's version of Math 125 is an introduction to algebraic number theory. Topics will include basic facts about algebraic number fields and integers, the Dirichlet unit theorem, discriminants of algebraic number fields, factorization of ideals in rings of integers, and p-adic fields. There are two primary objectives for this class: to prepare graduate students for a qualifying exam in algebraic number theory, and to prepare students to read about class field theory on their own.

## Instructor information

Name: Andrew Yang
Office: Kemeny 316
Office Hours: By appointment

## Homework Assignments

Unless otherwise specified, homework problems refer to questions in Neukirch.

• HW4, due Friday, 6/03/2011: Chapter 2.1: #5, Chapter 2.2: #3, #5, #6, Chapter 2.3: #1, #2.
• HW3, due Wednesday, 5/25/2011: Chapter 1.7: #2, #3, #5, Chapter 1.8: #4, #8, Chapter 1.9: #3, Chapter 1.10: #1, #2, #4
• HW2, due Monday, 4/25/2011: Chapter 1.4, #1, 2, Chapter 1.5, #3, Chapter 1.6, 2, 3, 5.
• HW1, due Monday, 4/11/2011: Chapter 1.1, #1, 3, 5, Chapter 1.2, #4, 6, 7, Chapter 1.3, #2

Grading in this class will be based on homework assignments and in-class participation. Homework assignments will be provided from the text on a weekly basis. In-class participation will probably consist of presentation of selected homework problems, and some more extensive projects later in the class. This will either be presentations by people in the class on topics of their choosing, or some interactive activities using computational software (almost certainly using Sage).

## Textbook

The required book for this class is Algebraic Number Theory, by Jurgen Neukirch, ISBN 3540653996. The textbook is long, and we will only be covering something like 1/3 of it, but hopefully by the end of the class you will be equipped to read the rest of the book on your own. This book has more of an emphasis on drawing analogies with algebraic geometry than the typical algebraic number theory text.

There are many books written about algebraic number theory. The following is a list not meant to be all-inclusive, but if you are interested in learning about the subject from slightly different points of view consider reading the following books:

• Algebraic Number Theory, by Serge Lang. Not too long, and covers the standard curriculum, although sometimes Lang can be a bit confusing to read.
• Algebraic Number Theory, by Cassels and Frohlich. This book arose from a conference in algebraic number theory in 1965, so its chapters are written by a variety of different authors. One highlight is the inclusion of John Tate's Ph.D. thesis, which had never before been widely available in print.
• A Course in Arithmetic, by Jean-Pierre Serre. A classic, this book is very short, and is not a systematic treatment of algebraic number theory, but what it does contain (Hasse-Minkowski principle for quadratic forms, modular forms) is pure gold. You should definitely read this book!
• Algebraic number theory notes by J.S. Milne, available online at his website. Milne has a collection of well-written notes on many number theory topics, all available for free.
• A Course in Computational Algebraic Number Theory, by Henri Cohen. This book takes a different tack than a "typical" number theory textbook, focusing not just on theory but also on the computational aspects of the subject. Cohen is a world leader in computational number theory, and was one of the original authors of the number theory package PARI.