Modern Algebra I

Math 627A
Course number: 21770
Fall 2008
Meeting MWF 11:00-11:50
San Diego State University
Final Exam: Mon. Dec. 15, 10:30-12:30

Professor: Mike O'Sullivan
Web page:
Office: GMCS #579, ext. 594-6697
Office Hours: MWF 8-9, MW 1-2,
                             You may also make an appointment for another time or stop by my office. If I am in and available, we can talk.

Notes on group theory.

Detailed Information


Course Description

The ultimate goal of this course is to introduce the main theorems and standard examples in Galois theory. Along the way we will cover the fundamentals of groups, commutative rings, and fields.

The roots of Galois theory lead back to problems posed by the ancient Greeks and their predecessors. Greek geometers achieved remarkable constructions with ruler and compass, but a number of simple, nagging, problems remained unresolved until the Rennaisance. For example: Is it possible to trisect an arbitrary angle? Which regular polygons are constructible? In algebra, several civilizations investigated the solution of a quadratic equation (see MathWorld article ). The attempt to find a solution for higher degree equations was another project that occupied numerous mathematicians. The resolution of these ancient questions culminated in Galois' theory of fields. It is a delightful subject, and the modern treatment highlights the interplay between three key areas of algebra: groups, rings and fields.


Ash, Robert, Abstract Algebra: The basic graduate year . available online

This book is a concise and direct treatment of the fundamentals of graduate level algebra.

Hungerford, Abstract Algebra: An Introduction 2nd ed.

This has been the standard text for undergraduate algebra at SDSU for a few years. It will be useful for review of material that is covered tersely in Ash's book.


A good understanding of the basics of groups, rings and fields (Math 521A and 521B is plenty). I will assume you are conversant with the following material, and need only a gentle reminder. I suggest you review the main points in the sections from Hungerford noted below.

Foundational Topics

Some of the following topics will be familiar from your undergraduate course. We will cover them in greater depth, and with more attention to details. I expect this material to take 6-9 weeks.

Galois Theory

The main topics are


There will be several (6-8) homework assignments a midterm and a final exam. The final grade will be weighted as follows.
Problem Sets 35%
Midterm 35%
Final Exam 30%