MA20310 - INTRODUCTION TO ABSTRACT ALGEBRA

Module Identifier	MA20310
Module Title	INTRODUCTION TO ABSTRACT ALGEBRA
Academic Year	2002/2003
Co-ordinator	Dr T McDonough
Semester	Semester 1
Pre-Requisite	MA11010
Mutually Exclusive	MX30310
Course delivery	Lecture	19 x 1 hour lectures
	Seminars / Tutorials	3 x 1 hour example classes
Assessment	Semester Exam	2 Hours (written examination)	100%
	Supplementary Exam	2 Hours (written examination)	100%

Learning outcomes

On completion of this module, students should be able to:

determine whether given relations are equivalence relations;
apply the division algorithm in a range of contexts;
apply the Euclidean algorithm to determine highest common factors in appropriate systems;
perform computations using 'modulo' arithmetic;
describe constructions of number systems using equivalence relations;
determine whether given algebraic systems are groups;
determine whether given subsets of groups are subgroups;
determine whether elements and subgroups of a group possess a variety of properties;
state and apply the fundamental homomorphism theorems;
prove propositions concerning numbers, polynomials and groups.

Brief description

In this module, properties of the integers and the polynomials with number coefficients in a single variable are developed in a formal setting. This allows similar properties for other number systems to be inferred from the formal propositions. It also points the way towards the construction of number systems with other desirable properties. The axiomatic approach is then used to develop many of the elementary propositions arising in the context of groups -- algebraic systems which have a simple formal description and which occur naturally as descriptors of symmetry, both within mathematics and outside of it.

Aims

To provide an introduction to abstract algebra by studying the basic structure systems of integers and polynomials, by constructing other related number systems and by developing the elementary aspects of theory of groups. To show how a variety of systems, from disparate areas, may be dealt with in a unified way by the development of an abstract theory which embraces them.

Content

1. SETS AND MAPPINGS
Review of basic concepts. Cartesian products. Composition of mappings -- associativity. Binary operations. Equivalence relations.
2. PRINCIPLES AND METHODS OF FORMAL PROOF
Deduction rules. Agruments by contradiction. The natural numbers. Well-ordering and induction.
3. NUMBERS AND POLYNOMIALS
Factors. Division and Euclidean algorithms. Primes and irreducibles. The Fundamental Theorem of Arithmetic. Uniqueness of factorisation of polynomials. Congruences. 'Modulo' arithmetic. Solution of linear congruences in one unknown. Some Classical congruences.
4. NUMBER SYSTEMS
Constructions using equivalence relations. Quadratic extension fields.
5. GROUPS
The group concept. Formal verification of the group axioms in a selection of examples. The direct product construction. Subgroups -- the subgroup criterion. Matrix groups, plane symmetry groups. Cosets of a subgroup. Index of a subgroup. Lagrange's Theorem. Homomorphism and isomorphism of groups.

Transferable skills

Skills in the use and analysis of numerical information, analytical reasoning, writing in an academic context and self-management are developed with regular written assignments completed in the student's own time by given deadlines.

Reading Lists

Books
D A R Wallace. (1998) Groups, Rings and Fields. Springer 3540761772
T A Whitelaw. (1995) An Introduction to Abstract Algebra. 3/e. Chapman and Hall 0751401471
N H McCoy and G J Janusz. (2001) Introduction to Abstract Algebra. 6/e. Harcourt/Academic Press 0123803926
J B Fraleigh. (1999) A First Course in Abstract Algebra. 6/e. Addison-Wesley 0201474360
R B J T Allenby. (1991) Rings, Fields and Groups. 2/e. Edward Arnold 0340544406