Modern Algebra I

Math 627A
Course number: 21604
Fall 2005
Meeting MW 2:00-3:15
GMCS #307
San Diego State University
Final Exam: Wed. Dec. 21, 1:00-3:00

Professor: Mike O'Sullivan
Web page: http://www.rohan.sdsu.edu/~mosulliv
Email: mosulliv@math.sdsu.edu
Office: GMCS #579, ext. 594-6697
Office Hours: M: 1:00-1:30, Tu: 2:00-3:30, W: 1-1:30, 2:15-3:30, Th: 2:00-3:30.
Other times: by appointment.

Course Description

We will study Galois 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.

After a brief introduction to the classical problems, we will start with fields and field extensions. This leads naturally to a study of commutative rings which are necessary to construct new fields from the simplest ones. Finally we will see that automorphisms of a field are useful for analyzing field extensions. This leads to group theory, in particular normal subgroups and solvable groups. Galois great theorem, which is the culmination of the course, establishes a correspondence between field extensions and groups of automorphisms.

Required Materials

Rotman, Joseph Galois Theory Springer-Verlag, Universitext, 1990.

This book is a concise and direct treatment of the fundamentals. The exercises are an integral part of the text: many of the simpler or more routine proofs as well as some important theorems are left to the exercises. You should plan to read and at least sketch solutions to most of them. I will also cover some of the material in my lectures.

Prerequisites

A good understanding of the basics of groups, rings and fields (Math 521A and 521B is plenty). In particular you might want to review the following.
• The integers: The division theorem, greatest common divisor, Euclidean algorithm, prime numbers, unique factorization.
• Polynomial rings in one variable over a field (e.g. the rationals): The division theorem, greatest common divisor, Euclidean algorithm, irreducible polynomials, the correspondence between factors of a polynomial and its roots, unique factorization.
• Commutative rings: We will only treat commutative rings, and they will usually be derived from the integers or a polynomial ring over the rationals. You should be familiar with ideals, homomorphisms (ring maps), the quotient of a ring by an ideal and the field of quotients of an ideal. We will review and then explore this material in more depth.
• Linear Algebra: Vector space over a field, nullspace, subspace, dimension, linear independence, spanning set, basis.