Aims

The aim of the module is:

• To show how unified mathematics is by using ideas from different modules.
• To show how very abstract ideas can be used to derive concrete results.
• To introduce students to one of the high points of 19th century algebra.

Learning objectives

At the end of the module you should be able to:

• construct fields by quotienting polynomial rings by maximal ideals;
• know Eisenstein's irreducibility criterion;
• know that any irreducible polynomial has a suitable extension field;
• know what the degree of a field extension is and that it is multiplicative;
• know about the form of the elements in a simple extension;
• know what a splitting field of a polynomial is;
• know about the allowable operations with straightedge and compasses;
• know that the constructible real numbers form a field which is closed under extraction of square roots;
• understand that each constructible real number lies in a field of degree a power of two over Q and the consequences of that fact;
• understand some of the results about automorphisms of an extension field and the fixed field of a group of automorphisms;
• know the statement of the Galois Correspondence Theorem and be able to apply it to straightforward examples;
• understand some of the consequences of the Galois Correspondence.

Syllabus

Recommended texts

• E Artin, Galois Theory, University of Notre Dame Press (S 2.82 ART).
• H M Edwards, Galois Theory, Springer (S 2.82 EDW).
• J Rotman, Galois Theory, Springer (S 2.82 ROT).
• I Stewart, Galois Theory, Chapman and Hall (S 2.82 STE).

Teaching

• Spring Term
• 2 lectures per week
• 1 problems class per week

Assessment

One and a half hour closed examination in Summer Term (90%),
Coursework (10%). Note that coursework submitted after the advertised deadlines will be given a mark of zero.

Elective information

This module is not available as an elective.

Prerequisites

For students under the old scheme, if taken before 2010/11:

• Foundations Taught in: Core Mathematics
• all Taught in: Introduction to Group Theory 
• all Taught in: Linear Algebra 
• all Taught in: Groups and Rings

For students under the new scheme, if taken from 2010/11:

• Calculus MAT00001C (30 credits)
• Introduction to Group Theory MAT00006I (10 credits)
• Linear Algebra MAT00008I (20 credits)
• Groups Rings and Fields MAT00005I (20 credits)