|| MX30310 |
|| INTRODUCTION TO ABSTRACT ALGEBRA |
|| 2003/2004 |
|| Dr T McDonough |
|| Semester 1 |
|| MA11010 |
|| MA20310 |
| Course delivery
|| Lecture || 19 x 1 hour lectures |
|| Seminars / Tutorials || 3 x 1 hour example classes |
|Assessment Type||Assessment Length/Details||Proportion|
|Semester Exam||2 Hours (written examination) ||100%|
|Supplementary Exam||2 Hours (written examination) ||100%|
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;
prove propositions concerning numbers, polynomials and rings;
state and apply the fundamental homomorphism theorems.
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 some of the elementary propositions arising in the context of rings -- algebraic systems which generalise the number systems already encountered.
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 rings. 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.
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.
The ring concept. Axiomatic definitions and elementary deductions from the axioms. Homomorphism and isomorphism of rings. Subrings, ideals and factor rings.
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.
D A R Wallace (1998) Groups, Rings and Fields
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
This module is at CQFW Level 6