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MAT-40016 - Module Specification
School of Computing and Mathematics
Faculty of Natural Sciences

For academic year: 2022/23 Last Updated: 19 January 2023

MAT-40016 - Field and Galois Theory
Coordinator: Paul 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 2022/23

None

Available as a Free Standing Elective

No

Co-requisites

None

Prerequisites

MAT-20025 Abstract Algebra

Barred Combinations

None

Description for 2022/23

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: Exam weighted 100%
3 hour examination
A closed book examination assessing students' fluency in the concepts and ideas introduced in the module and their ability to apply these ideas to solve "seen similar" and unseen problems.

Please note that for the 2022/23 Academic Year programmes may be adapted to operate in accordance with a flexible, digital approach. This means that there may be some changes made to how modules are delivered and assessed. The latest information will be provided to you by your School in module handbooks. We endeavour to give you as much notice as possible of any changes and details about the circumstance in which changes are made to programmes can be found in the Student Agreement, available at: https://www.keele.ac.uk/student-terms-and-conditions/

Last Updated: 19 January 2023