# Galois Theory

## Unit aims

To present an introduction to Galois theory in the context of arbitrary field extensions and apply it to a number of historically important mathematical problems.

## Unit description

After reviewing some basic properties of polynomial rings, we will introduce the basic objects of study: field extensions and the automorphism groups associated to them. We will discuss certain desirable properties for field extensions and then demonstrate the fundamental Galois correspondence. This will be used to analyse some specific polynomials and in particular to exhibit a quintic which is not soluble by radicals. We will end with applications to finite fields and to the fundamental theorem of algebra.

## Relation to other units

This is one of three Level 7 units which develop group theory in various directions. The others are *Representation Theory* and *Algebraic Topology*.

## Learning objectives

To gain an understanding and appreciation of Galois theory and its most important applications. To be able to use the theory in specific examples.

## Transferable skills

Using an abstract framework to better understand how to attack a concrete problem.

## Syllabus

Review of field extensions including degree of a field extension, algebraic elements, ruler and compass constructions. Extending field homomorphisms and the Galois group of an extension. Algebraic closures. Splitting field extensions. Normal extensions. Separability. Inseparable polynomials, differentiation, and the Frobenius map. Simple extensions and the Primitive Element Theorem. Fixed fields and Galois extensions. The main theorems of Galois theory. Finite fields. Solvability by radicals: quadratic, cubic, and quartic polynomials. Cyclotomic polynomials and cyclotomic extensions.

Some possible additional topics, if time permits:

General polynomials. Construction of regular polygons. Cyclic extensions and Abel's theorem. The inverse Galois problem. Hilbert's 13th Problem.

## Reading and References

Recommended Text:

D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, 1986.

See also:

Ian Stewart, Galois Theory, 3rd ed. Chapman & Hall, 2003;

Emil Artin, Galois Theory, New ed. Dover, 1998.

**Unit code:** MATH M2700 Galois Theory

**Level of study:** M/7

**Credit points:** 20 credit points

**Teaching block (weeks):** 2 (13-24)

**Lecturer:** Dr Lynne Walling

## Pre-requisites

MATH33300 Group Theory, MATH21800 Algebra 2

## Co-requisites

None

## Methods of teaching

Lectures and exercises.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 90% from a 2 hour 30 minute exam
- 10% from selected homework questions

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.