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.
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.
- 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).
- Spring Term
- 2 lectures per week
- 1 problems class per week
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.
This module is not available as an elective.
For students under the old scheme, if taken before 2010/11:
For students under the new scheme, if taken from 2010/11: