Modern Algebra I

Math 627A
Course number: 21833
Fall 2012
Meeting TuTh 5:30-6:45
GMCS 328
San Diego State University
Last class: Thurs. Dec 6.
Final Exam: Thurs. Dec. 13, 3:30 - 5:30.

Professor: Mike O'Sullivan
Web page:
Office: GMCS #579, ext. 594-6697
Office Hours: Tu Th, 1:00-2:30.
                             You may make an appointment for another time, or just stop by my office. If I am in and available, we can talk.

Detailed Information

Review for first exam
My lecture notes
Lecture notes on group isomorphism theorems
Lecture notes on field extensions

Course Description

The goal of this course is to introduce the main theorems and standard examples in Galois theory. Along the way we cover the fundamental theory of groups and fields, and a bit of ring theory. Next semester, in 627B, the focus will be on ring theory.

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.

The course begins with a review of the fundamentals of each area, working with lots of examples. We start with the group axioms and basic properties: subgroups, homomorphisms and conjugation. We then introduce rings and fields, with a focus on constructing new fields by computing modulo an irreducible polynomial. We use the computer algera system Sage to explore complicated examples. 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 develops deeper results in group theory: quotient groups, the isomorphism theorems, the Sylow theorems, group actions, and solvability of a group. The third part of the course starts with the study of field extensions, and culminates with Galois' theorem connecting field extensions to automorphism groups. We apply the theory to as many examples and classical problems as time allows.



I will assume that you have some experience with groups, rings and fields (SDSU's Math 521A and 521B is plenty). Although we develop the theory from the axioms, you need some familiarity with the material in order to keep up with the pace. The most important prerequisite material is from our 521A course in which we cover the first 6 chapters of Hungerford's book. I *strongly* suggest that you review the main points in the sections noted below during the first weeks of the semester (or before it starts).

It is also worth reviewing the following topics.

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 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 roughly as follows.
Problem Sets 35%
Midterm 25%
Final Exam 40%