Unit information: Galois Theory in 2013/14

Unit name	Galois Theory
Unit code	MATHM2700
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Walling
Open unit status	Not open
Pre-requisites	MATH33300
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

After reviewing some basic properties of polynomials 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.

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.

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:

Solubility of general polynomial equations. Construction of regular polygons. The inverse Galois problem. Hilbert's 13th Problem.

Relation to Other Units

This is one of three Level 4 units which develop group theory in various directions. The others are Representation Theory and Algebraic Topology.

Intended Learning Outcomes

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.

Teaching Information

Lectures and exercises.

Assessment Information

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.

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.