Modern Algebra I

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 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.

Resources

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

  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 will write some lecture notes. 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.

It is also worth reviewing the following topics.

• Commutative rings (H 3.1-3, 6.1): We will only treat commutative rings, and they will usually be derived from the integers or a polynomial ring over the rationals.
• Linear Algebra: Vector space over a field, nullspace, subspace, dimension, linear independence, spanning set, basis.

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.

• Groups: Normal subgroups and the isomorphism theorems. Groups acting on sets, orbits and stabilizers. The Sylow theorems.
• Commutative Rings: Ideals, homomorphisms, and quotient rings of polynomial rings over a field. Criteria for irreducibility of polynomials.
• Fields: Field extensions. Adjoining a root of a polynomial. Splitting field of a polynomial.

Galois Theory

• Normal and separable field extensions.
• The Galois group of a field extension.
• The fundamental theorem: relating intermediate fields of K/F to subgroups of the Galois group of K/F
• Standard examples: finite fields, cyclotomic fields, solvability by radicals, constructible numbers.

Grading

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%