18.786 | Spring 2010 | Graduate

# Topics in Algebraic Number Theory

## Syllabus

### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Prerequisites

Real Analysis (18.100) and Algebra II (18.702).  A background in elementary number theory (e.g., 18.781) is strongly recommended.

### Overview

This course is an introduction to algebraic number theory. We will follow Samuel’s book Algebraic Theory of Numbers to start with, and later will switch to Milne’s notes on Class Field theory, and lecture notes for other topics.

There will be assigned readings for every class. I will go through the proofs of the more important theorems in class, and maybe some extra material (for instance, proofs omitted in the book).

There will also be a focus on computer algebra calculations, esp. using gp/Pari and sage.  Siman Wong’s gp/Pari tutorial is helpful.

Topics to be covered include

• Some basic commutative algebra
• Rings of integers, Dedekind domains, ideals and factorization
• Discriminants, ramification, structure of Galois groups
• Valuations, local fields
• Class groups, Dirichlet’s unit theorem
• Some basic theory of quadratic forms
• Class field theory

### Textbook

Samuel, Pierre. Algebraic Theory of Numbers. Translated by Allan J. Silberger. Mineola, NY: Dover, 2008. ISBN: 9780486466668.

Milne, J. S. Algebraic Number Theory, 2009. (Available at Mathematics Site).

There will be weekly problem sets. If you are an undergraduate or a first-year graduate student, I will assign you a grade based on homework and exams. Even otherwise, I strongly recommend doing the homework, to learn the material.

The breakdown of the grade is:

ACTIVITIES PERCENTAGES
Homework 50%
Midterm 20%
Take-home final 30%

As usual, you are encouraged to work on the homework in groups, but you must write up your own solutions, and I would like you to specify on your homework who was in your working group. On the take-home exam, you are to work on your own using only the specified resources (the book, your course notes, any book from the library, but not any human and not Google or Wikipedia).

Spring 2010
Problem Sets