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Galois Theory (MATH M2700)

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Administrative Information

  1. Unit number and title: MATH M2700 Galois Theory
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: This unit has been given from time to time for many years.
  6. Unit Organiser: Abhishek Saha
  7. Lecturer: Dr Abhishek Saha, Dr Sean Prendeville
  8. Teaching block: 1
  9. Prerequisites: MATH33300 Group Theory (MATH 21800 Algebra 2 is strongly recommended, and will be a formal prerequisite from 2013-14 onwards)

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.

General Description of the Unit

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.

Teaching Methods

Lectures and exercises.

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.

Assessment Methods

The assessment mark for Galois Theory is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Award of Credit Points

To gain credit points for this unit, students must

  • either gain a pass mark (50 or over) for the unit
  • or gain a mark of at least 30 for the unit, and make satisfactory attempts at selected questions on at least five homework assignments.

Transferable Skills

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

Texts

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.

Syllabus

Polynomial rings. Irreducible polynomials.
Field extensions. Algebraic and transcendental elements. Simple extensions.
Degree of an extension. Splitting fields. Algebraic closure.
Impossibility of some geometric constructions.
Fixed fields and Galois groups.
Splitting fields and normal extensions. Separable extensions.
The fundamental theorem of Galois theory.
Solutions of polynomials by radicals. Insolubility of a quintic.
Finite fields. Transcendental elements and algebraic independence. Applications. The fundamental theorem of algebra.

Some possible additional topics, if time permits:

General polynomials.
Construction of regular polygons.
Cyclic extensions over fields of positive characteristic.
The inverse Galois problem. Hilbert's 13th Problem.