## MA4GT-Galois Theory

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:7
Terms in which taught: Autumn / Spring term module
Pre-requisites: MA2AL2 Algebra II
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3A7 Galois Theory MA3GT Galois Theory
Module version for: 2017/8

Module Convenor: Dr Graham Williams

Summary module description:

A presentation of the Galois theory of polynomial equations,making full use of the powerful language and techniques of modern algebra to develop the results in an efficient and streamlined way. The course builds on the group, ring and field theory studied in Part 2, and shows the profound consequences which flow from what at first sight may appear as mere abstraction. Along the way the solution of some famous problems of classicalgeometry will also be presented.

Aims:

To illustrate the interconnections between diverse mathematical disciplines.To show how the development and intelligent use of mathematical tools can solve classical problems in an elegant and simple way.To give examples of how algebraic theory can successfully replace intractable manipulation.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

• understand and work with normal field extensions, and the connection between these and splitting fields;

• determine the normal closure of an extension;

• understand and be able to prove fundamental results such as Dedekind’s Lemma on linear independence of automorphisms;

• calculate Galois groups of various extensions and polynomials;

• prove basic results concerning solvable groups;

• be able to determine trisectability or otherwise of given angles;

• demonstrate that a given polynomial is not solvable by radicals.

This module will be assessed to a greater depth than the excluded module MA3GT.

Students will appreciate how the fusion of geometric reasoning with the developing methods of field theory in the 19th century resulted in the successful solution of several outstanding mathematical problems from antiquity.

Outline content:

The module mainly concentrates on the so called Classical Galois theory, which more or less is the original theory, but with present day notation and presentation. We shall discuss normal extensions and their Galois groups, and apply this to the problem of solvability of quintic and higherdegree polynomial equations. Since the apparatus of Galois theory lends itself to solving some of the famous problems of classical geometry (for example the impossibility of trisecting a general angle using only ruler and compasses) we shall also examine these problems in the course.

Brief description of teaching and learning methods:

Lectures supported by problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 20 Guided independent study 80 80 Total hours by term 100.00 100.00 Total hours for module 200.00

Summative Assessment Methods:
 Method Percentage Written exam 100

Other information on summative assessment:

One examination paper.

Formative assessment methods:

Problem sheets.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:

3 hours.

Requirements for a pass:

A mark of 50% overall.

Reassessment arrangements:

One examination paper of 3 hours duration in August/September.