Numbers & Rings (Ord) (U01646)

? Credit Points : 10 ? SCQF Level : 9 ? Acronym : MAT-3-NuRO

Syllabus summary: Factorisation theory of integers and polynomials in one variable over a field. Euclidean domains. Unique Factorisation Domains. Congruences and modular arithmetic. Ideals and quotient rings. Gauss's Lemma and the Eisenstein criterion for irreducibility of polynomials over the integers.

Entry Requirements

? Pre-requisites : MAT-2-FoC, MAT-2-SVC, MAT-2-LiA, MAT-2-MAM or MAT-2-am3, MAT-2-mm3, MAT-2-am4, MAT-2-mm4 or MAT-2-mi3, MAT-2-mi4

? Prohibited combinations : MAT-3-NuR

Subject Areas

Home subject area

Specialist Mathematics & Statistics (Ordinary), (School of Mathematics, Schedule P)

Delivery Information

? Normal year taken : 3rd year

? Delivery Period : Semester 2 (Blocks 3-4)

? Contact Teaching Time : 2 hour(s) per week for 11 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Monday	14:00	14:50	KB
Lecture	Thursday	14:00	14:50	KB

Summary of Intended Learning Outcomes

The following are the learning objectives for the Honours version, MAT-3-NuR; for this (Ordinary) version there is more emphasis on the technical, rather than conceptual elements, which will be reflected by a different examination.

1. To be able to use the division algorithm and euclidean algorithm in apppropraiate settings.
2. To be able to apply the Eisenstein criterion for irreducibility of integer polynomials.
3. To understand the necessity for rigorous proofs, as exemplified by the confusions due to assuming unique factorisation is universally applicable.
4. To understand the idea of defining operations on sets defined by equivalence relations and to understand the notion of 'well-defined' for such definitions.
5. To understand the abstract notions of ideals and factor rings and to be able to work with these notions in elementary situations.
6. Given an irreducible polynomial over a field, to be able to construct an extension field that contains a root of the polynomial.