Modern Algebra II

Detailed Information

Course Description

The 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 (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 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 an introduction to classical problems and the link with field theory. We will then spend a few days on fundamental properties of polynomial rings: irreducibility, derivatives, and some related topics (including the abc conjecture). After rings we move to more advanced group theory. We will start with concrete constructions of groups: groups from rings, automorphism groups, semi-direct products, group extensions, group representations. Then we develop more theory: group actions, the Sylow theorems, composition series, solvable groups. At some point we return to field theory: the classification and structure of finite fields, the standard non-separable extension, normal and Galois extensions. We finish with Galois' theorem connecting field extensions to automorphism groups. We apply the theory to as many examples and classical problems as time allows.

Resources

• Ash, Robert, Abstract Algebra: The basic graduate year .

  This text is a concise and direct treatment of the fundamentals of graduate level algebra. Since it is a bit too concise in some areas I wrote some lecture notes.

• O'Sullivan, Michael E. Lecture Notes for 627B

  I hope to add to these notes as the semester progresses. I also suggest having a copy of a good undergraduate level text to have access to more detailed explanations and lots of examples. Three such examples are below.

• Hungerford, Abstract Algebra: An Introduction 2nd ed.

  This has been the standard text for undergraduate algebra at SDSU for a few years.

• Gallian, Contemporary Abstract Algebra.

  A widely used and well written text.

• Judson and Beezer, Abstract Algebra: Theory and Applications

  This is another good undergraduate algebra text. Available free online. See the button for "Sage and AATA." Download "Sage Worksheet Collection." Open the zip file from the sage notebook.

• SDSU Sage Tutorial Updated 1/28/12. There may be updates of the tutorial during the semester.

Prerequisites

• The integers (H 1.1-3, 2.1-2): The division theorem, greatest common divisor, Euclidean algorithm, prime numbers, unique factorization, modular arithmetic.
• Polynomial rings in one variable over a field (e.g. the rationals) (H. 4.1-4, 5.1-2): The division theorem, greatest common divisor, Euclidean algorithm, irreducible polynomials, unique factorization. The correspondence between linear factors of a polynomial and roots. Congruence modulo a polynomial.
• Linear Algebra: Vector space over a field, nullspace, subspace, dimension, linear independence, spanning set, basis.