# Math 413 syllabus

Abstract Algebra I (W)

Bulletin description:
An introduction to group theory and ring theory. Topics

include permutations and symmetrics, subgroups, quotient groups,

homomophisms, as well as examples of rings, integral domains, and fields.

Prerequisites:   MA 237 and a "C" or better in one of the following: MA 311, 320, 334. Also needed: a "C' or better in EH 102 or EH 105.

Text:   A Book of Abstract Algebra Pinter Second Edition Dover Publishers ISBN: 0-486-47417-8

Learning Objectives:

The goal of this course is to familiarize students with some fundamental algebraic structures (groups, rings, fields) and their applications. Mathematical exposition is emphasized, and students are expected to understand and construct proofs.  By presenting a variety of accessible topics, the course not only gives an overview of the nature and utility of algebra but hopefully will also show some of its beauty.  Students who successfully complete MA 5413 will be able to:

Write short proofs (direct, by contradiction, and using the contrapositive)

Disprove algebraic statements by finding a counterexample

State, justify, and apply basic properties of groups, rings, and fields.

Verify that a given subset of a group is a (normal) subgroup

Verify that a given subset of a ring is a subring (ideal)

Find the order of a given group element

Prove that two groups or rings are isomorphic or are not isomorphic

Find the characteristic of an integral domain

Construct the field of quotients of an integral domain

State and prove Cayley’s Theorem, the structure theorem of cyclic groups,

the subgroup theorem for cyclic groups, Lagrange’s Theorem, the

fundamental homomorphism theorems for groups and rings, the

isomorphism theorems for groups and rings, the correspondence

theorems for subgroups, normal subgroups, subrings, ideals, and

Cauchy’s Theorem.

Last updated February 12, 2014