#### Description

This course discusses the basic elementary facts about the integers, groups, rings, integral domains, and fields. It is recommended strongly for all students planning to go to graduate school in mathemetics. Credit will only be given for one of Math 402 and 403.

#### Prerequisites

C- or better in (MATH 240, MATH 241, and MATH 310) or (MATH 340, MATH 341, and MATH310)

#### Topics

**Groups**

Definition and examples

Subgroups

Normal subgroups and quotient groups

Homomorphisms

Isomorphism theorems

Cyclic groups

Permutation groups, Cayley's Theorem

**Rings**

Definition and examples

Polynomial rings

Ideals and quotient rings

Integral domains

Euclidean domains, unique factorization

Prime ideals, maximal ideals

**Fields**

Quotient fields

Extensions

Algebraic extensions

Impossibility of trisection of angle,