This module is not currently running in the selected year. The information shown below is for the academic year that the module was last running in, prior to the year selected.

2011/12 Undergraduate Module Catalogue

MATH2033 Rings, Polynomials and Fields

10 creditsClass Size: 90

Module manager: Professor P Martin
Email: ppmartin@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2011/12

Pre-requisites

MATH1022

Introductory Group Theory

This module is approved as an Elective

Module summary

A ring is an algebraic system in which addition, subtraction and multiplication may be performed. Integers, polynomials and matrices all provide examples of rings, so this notion covers an important range of mathematical structures. They are studied in this module.One topic is the generalization to some other rings of the Fundamental Theorem of Arithmetic, that every positive integer can be written in a unique way as a product of primes. Another topic is Kronecker's theorem that every non-constant polynomial with coefficients in a field has a root in some larger field. Combined, one can begin to understand the possible finite fields.

Objectives

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

a) define some of the main concepts about rings, polynomials and fields
b) state and prove some of the basic results about rings, polynomials and fields
c) state the axioms of a ring and deduce basic properties directly from them
d) identify subrings, ideals and units in the main examples of rings
e) use the First Isomorphism Theorem to exhibit isomorphisms between rings
f) demonstrate understanding of unique factorisation or lack of it
g) identify irreducibles in various examples of rings, using appropriate tests
h) make computations in fields obtained by adjoining an algebraic element

Syllabus

1. Rings, fields and integral domains. Units. Subrings. Examples, such as the ring obtained by adjoining a square root to the integers, matrix rings, polynomial rings and the field of fractions of an integral domain.
2. Ideals, factor rings, homomorphisms and isomorphisms. Principal ideal domains. The main examples of principal ideal domains. The First Isomorphism Theorem for rings.
3. Factorization in integral domains. The notion of a greatest common divisor (at least in a principal ideal domain). Euclid's Algorithm. Prime and irreducible elements in integral domains. Any principal ideal domain is a unique factorization domain. Fundamental Theorem of Arithmetic. Factorization of polynomials. Gauss' Lemma. Eisenstein's criterion. The rational root test.
4. Field extensions. Minimal polynomials of algebraic elements in field extensions. Structure of the field obtained by adjoining an algebraic element. Kronecker's Theorem. The characteristic of a field and the prime subfield. The number of elements in a finite field is a prime power. Examples of finite fields.

Teaching methods

Delivery type	Number	Length hours	Student hours
Workshop	10	1.00	10.00
Lecture	22	1.00	22.00
Private study hours			68.00
Total Contact hours			32.00
Total hours (100hr per 10 credits)			100.00

Opportunities for Formative Feedback

Examples sheets.

Methods of assessment

Coursework

Assessment type	Notes	% of formal assessment
In-course Assessment	.	15.00
Total percentage (Assessment Coursework)		15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 00 mins	85.00
Total percentage (Assessment Exams)		85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 27/02/2012

Disclaimer

Browse Other Catalogues

• Undergraduate module catalogue
• Taught Postgraduate module catalogue
• Undergraduate programme catalogue
• Taught Postgraduate programme catalogue

Errors, omissions, failed links etc should be notified to the Catalogue Team.PROD