MAT-40016 - Field and Galois Theory
Coordinator: Paul J Truman Room: MAC2.16 Tel: +44 1782 7 33246
Lecture Time: See Timetable...
Level: Level 7
Credits: 20
Study Hours: 200
School Office: 01782 733075

Programme/Approved Electives for 2021/22

None

Available as a Free Standing Elective

No

Co-requisites

None

Prerequisites

MAT-20025 Abstract Algebra

Barred Combinations

None

Description for 2021/22

This module gives an introduction to fundamental topics and concepts in modern abstract algebra via the systems of rings and fields. The highlight of the course is the study of Galois theory - a topic in which field theory and group theory come together to answer some of the fundamental questions in mathematics about polynomials and their roots. In particular, Galois theory allows us to prove that there is no general formula for finding the roots of a quintic polynomial. Several applications of the theory will also be given.

Aims
This module will provide students with a firm grounding in modern abstract algebra. The module will highlight Galois theory which is an extremely important discovery in the history of Pure Mathematics and shows the relevance of the theory to a variety of explicit problems.

Intended Learning Outcomes

establish and prove properties of rings and fields directly from the set of axioms: establish that a given polynomial is irreducible over a given base field using a variety of methods: prove that a given field extension is a Galois extension: appraise a finite Galois extension and create its Galois Group: evaluate and apply the Galois Correspondence to analyse finite field extensions:

Study hours

48 Hours lectures/classes
3 Hour final examination
149 Hours private study, reading, problem solving.

School Rules

None

Description of Module Assessment

1: Open Book Examination weighted 100%
3 hour open-book examination
An online examination combining "seen similar" and unseen material.