Galois Theory and Related Topics

Detailed Information

Course Description

The main goal of this course is to introduce the main theorems and standard examples in Galois theory. Along the way we will do some more advanced group theory (group actions, Sylow theorems, composition series, group extensions). We will focus on interesting examples of the Galois correspondence for finite fields, number fields and function 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 Renaissance. 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. 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.

A second goal of this course is to develop some of the main results for multivariate polynomial rings over a field. Fundamental theorems for univariate polynomial rings over a field ( k[x] ) parallel those for the integers: the quotient remainder theorem, the greatest common divisor theorem, the Euclidean algorithm, unique factorization. Interestingly, k[x] has additional properties that make some problems more tractable over k[x] than they are over Z. We will explore some extensions of the fundamental theorems, with appropriate modification, to multivariate polynomial rings over a field.

Course Materials

O'Sullivan: Groups Rings and Fields (Course Notes for Math 620 as background material).

O'Sullivan: Supplementary lecture notes to be supplied during the course.

Thomas Hungerford: Abstract Algebra: An Introduction. This is the textbook we use for our undergraduate algebra courses. It is well written and should be a nice reference to esh out details and give examples.

David Dummit, Richard Foote, Abstract Algebra. A massive standard reference for graduate level algebra, full of examples and detailed proofs.

Robert Ash: Abstract Algebra: The Basic Graduate Year. Available for free at https://web.archive.org/web/20230326022245/https://faculty.math.illinois. edu/~r-ash/Algebra.html Physical copies are published by Dover and are cheap.

William A. Stein et al. Sage Mathematics Software The Sage Development Team, 2011, http:www.sagemath.org

The Sage Reference Manual

Sage is an open source mathematics software package that incorporates numerous other open-source packages into a unified package. The Sage tutorial will help you get started.

Prerequisites

• Properties of the Integers: The division theorem and divisibility, the Euclidean algorithm, unique factorization, modular arithmetic.
• Polynomial Ring in One Variable: The division theorem, greatest common divisor, the Euclidean algorithm, unique factorization. The correspondence between factors and roots. Polynomial rings modulo a polynomial.
• Commutative Rings and ideals: The general language of rings and ideals. Integral domains, the quotient of an integral domain by an ideal, homomorphisms.
• Linear Algebra: nullspace, subspace, dimension, basis.

Format