School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2021/22 Last Updated: 23 September 2021

MAT-40016 - Field and Galois Theory

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MAT-20025 Abstract Algebra

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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.

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.

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:

48 Hours lectures/classes

3 Hour final examination

149 Hours private study, reading, problem solving.

3 Hour final examination

149 Hours private study, reading, problem solving.

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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.