Galois Theory - 2019 entry

Drawing on key ideas in the theory of groups and fields, you will learn core elements of the theory of field extensions.  You are already familiar with the idea that the real numbers can be extended to the complex numbers by introducing a new number as the square root of -1; Galois theory formalises such constructions and explores the intriguing relationship between groups and field extensions.

As an important application of Galois theory, you will understand why there can be no algebraic solution to the general quintic polynomial with rational coefficients.

Prerequisite module: MTH2002 Algebra, or equivalent

The aim of this module is to motivate and develop Galois theory both as an abstract theory and through the study of important applications.

On successful completion of this module, you should be able to:

 Module Specific Skills and Knowledge:

1. State and apply key definitions in Galois theory;

2. State, prove and apply core theorems in Galois theory.

Discipline Specific Skills and Knowledge:

3. Perform computations accurately;

4. Use abstract reasoning to solve a range of problems.

Personal and Key Transferable / Employment Skills and Knowledge:

5. Communicate your findings effectively in writing;

6. Work independently and manage your time and resources effectively.

- Review of the field axioms, the characteristic of a field, examples. Field extensions, degree, finite and algebraic extensions, extensions obtained by adjoining a root of an irreducible polynomial, degree in a tower of extensions; irreducibility criteria for Polynomials: Gauss' Lemma and Eisenstein's criterion.

- Splitting fields and algebraic closure. Separable and inseparable extensions. Cyclotomic polynomials and extensions. Automorphisms of a field. The group of automorphisms, the fixed field of a subgroup of automorphisms, the Galois correspondence. The fundamental theorem of Galois theory. Finite fields. Finite extensions of finite fields. The Galois theory of finite fields. Composite extensions and simple extensions. The primitive element theorem;

- Cyclotomic extensions and abelian extensions over Q. Abelian groups as Galois groups over Q. Cyclic extensions and Kummer theory. Galois groups of polynomials. Solvable and radical extensions: solution of cubic and quartic equations by radicals, insolvability of the quintic. Computation of Galois groups over Q. Hilbert's irreducibility theorem. Polynomials with Galois groups Sn and An.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

ELE: http://vle.exeter.ac.uk/

Reading list for this module: