#### Description

This is the less theoretical of the two upper level undergraduate abstract algebra courses. (Student's planning graduate work in mathematics should take the other course in abstract algebra, MATH 403.) The course covers the basics of groups, rings, integral domains and fields. There will be a detailed study of several groups, and a study of the properties of integers and polynomials. Emphasis is on the origin of the mathematical ideas studied and the logical structure of the subject. (This course is not open to mathematics graduate students. Credit will only be given for one of Math 402 and 403.) The last item in the list of topics could be replaced by other topics in algebra, for example, coding theory or crystallographic groups.

#### Prerequisites

MATH 240 or MATH 461 or MATH 341

**Topics**

**Preliminaries**

Induction

Equivalence relations

Functions

**Groups**

Groups and symmetries

Finite groups

Cyclic groups

Subgroups

Permutation groups

Cosets and Lagrange's Theorem

External direct product

Normal subgroups and factor groups

Homomorphisms

The structure of finite abelian groups

**Rings**

Introduction to rings

Integral domains

Ideals and factor rings

Ring homomorphisms

**Polynomial Rings**

Division Algorithm

Unique Factorization of Polynomials

Divisibility in Integral Domains

**Fields**

Extension Fields

Algebraic Extensions

Finite Fields

**Geometric Constructions**

Constructible numbers

Trisecting an angle

Duplicating a cube

Squaring a circle