CATALOG DESCRIPTION:

1701.340 Modern Algebra I                                                                                             3 s.h.
(Prerequisites: 1701.210 Linear Algebra & 1509.130 Intro. to Symbolic Logic)

This course includes: the natural numbers, integers, rationals, and reals as mathematical systems, and the introductory theory of groups, rings, integral domains, and fields.  Also included are homomorphisms and isomorphisms, subgroups, kernels, rings and ideals, and polynomial rings. At the option of the instructor, computer use can be required.
 

OBJECTIVES:

This course is designed to begin the student's development in the area of abstract algebra and mathematical reasoning.

 1. Introduction

  1.1 Logic, sets, relations, functions
  1.2 Equivalence relations, equivalence classes

 2. Groups

  2.1 Definition and examples
  2.2 Basic group theorems
  2.3 Cyclic groups
  2.4 Subgroups, normal subgroups
  2.5 Quotient groups
  2.6 Homomorphisms, isomorphisms
  2.7 Internal and External Direct Products

 3. Rings

  3.1 Definition and examples
  3.2 Ring theorems
  3.3 Homomorphisms
  3.4 Ideals
  3.5 Integral Domains, Division Rings, Fields
  3.6 Polynomials

4. Integral Domains

  4.1 Definition and examples
  4.2 Ordered integral domains
  4.3 Congruence and residue domains
 

TEXTS:

Rotman, J.J.  A FIRST COURSE IN ABSTRACT ALGEBRA, Prentice Hall (7th ed)
Pinter, C.C.     ABOOK OF ABSTRACT ALGEBRA, (2nd ed) MacCraw Hill, 1990.
Herstein, I.N. TOPICS IN ALGEBRA, 2nd. Ed., Xerox, Mass., 1975.
Herstein, I.N.  and Kaplansky, I. MATTERS MATHEMATICAL, Harper and Row, NY, 1974.