Modern Algebra II

Math 627B
Course number: 21784
Spring 2012
Meeting TuTh 5:30-6:45
GMCS 308
San Diego State University
Last class: Tues. May 8.
Final Exam: Tue. May. 15, 3:30-5:30

Professor: Mike O'Sullivan
Web page:
Office: GMCS #579, ext. 594-6697
Office Hours: Tu Th, 11:00-1:00.
                             You may also make an appointment for another time or stop by my office. If I am in and available, we can talk.

Detailed Information

My lecture notes
Lecture notes on group isomorphism theorems
Lecture notes on Galois' main theorem and supporting topics
Review for test.
Final Exam Preparation.

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 fundamental theory of groups, 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.

We will start with a review of the group axioms, basic properties, subgroups and homomorphisms by working with lots of examples. We will do the same for rings and fields, with a focus on constructing new fields by computing modulo an irreducible polynomial. The first part of the course ends with an exploration of the connection between solving polynomials and field extensions.

The second part of the course will develop the key results of group theory: quotient groups, the isomorphism theorems, the Sylow theorems, group actions, and solvability of a group. The third part of the course will start with the study of field extensions, culminating with Galois's theorem connecting field extensions to automorphism groups. We will then apply the theory to as many examples and classical problems as time allows.



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 30%
Midterm 30%
Final Exam 40%