- Dr Karl M Schmidt (Reader - Cardiff University)
|Delivery Type||Delivery length / details|
|Lecture||33 x 1 Hour Lectures|
|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:
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.
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.
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.
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. Linear congruences.
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 to non-commutativity, basic properties of quaternions.
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.
This module is at CQFW Level 5