# Module Information

Module Identifier
MX30310
Module Title
Introduction to Abstract Algebra
2015/2016
Co-ordinator
Semester
Semester 1
Mutually Exclusive
Pre-Requisite
Other Staff

#### Course Delivery

Delivery Type Delivery length / details
Lecture 22 x 1 Hour Lectures

#### Assessment

Assessment Type Assessment length / details Proportion
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:
1. determine whether binary operations satisfy various properties (e.g. associativity, distributivity, existence of identities and inverses);
2. determine whether given relations are equivalence relations;
3. apply the division algorithm in a range of contexts;
4. apply the Euclidean algorithm to determine highest common factors in appropriate systems;
5. perform computations using modulo arithmetic;
6. describe constructions of number systems using equivalence relations;
7. prove and apply propositions concerning numbers, polynomials and rings.

### Brief description

In this module, properties of the integers and the polynomials with number coefficients in a single variable are studied in a formal setting. Using equivalence relations, algebras of equivalence classes are constructed with many of the properties of the integers and the polynomials. The notion of a ring emcompasses all the algebras encountered. The axiomatic approach is then used to establish elementary propositions for all rings and to provide a general context for the constructions.

### 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 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.

### Content

1. SETS AND MAPPINGS: Review of basic concepts. Cartesian products. Composition of mappings -- associativity. Binary operations. Distributivity. Equivalence relations.
2. THE INTEGERS: Factors. Division and Euclidean algorithms. Primes. Units. The Fundamental Theorem of Arithmetic.
3. POLYNOMIALS: Factors. The Remainder Theorem. Division and Euclidean algorithms. Irreducibles. Units. Uniqueness of factorisation of polynomials.
4. ARITHMETIC MODULO n: The congruence relation modulo n. Congruence classes. The algebra of classes, Z_n. Units and irreducibles.
5. POLYNOMIALS MODULO p(x): The equivalence relation modulo p(x). The equivalence classes. The algebra of classes, F[x]_p(x). Units and irreducibles. Finite fields.
6. RINGS: The ring concept. Axiomatic definitions and elementary deductions from the axioms. Homomorphism and isomorphism of rings. Ideals and factor rings. The homomorphism theorem.
7. QUATERNIONS: Introduction and basic properties of quaternions.

### 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.

### Notes

This module is at CQFW Level 6