MA3GAL-Galois Theory

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:6
Terms in which taught: Autumn term module
Pre-requisites: MA1FM Foundations of Mathematics or MA0FMNU Foundations of Mathematics and MA1LA Linear Algebra or MA1LANU Linear Algebra and MA2ALA Algebra or MA2ALANU Algebra
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2023/4

Module Convenor: Dr Basil Corbas
Email: b.corbas@reading.ac.uk

Type of module:

Summary module description:

Galois Theory examines the conditions under which polynomial equations can be solved using elementary algebraic operations. From the educational point of view it is the missing link between school algebra and so called abstract algebra. It unravels astonishing connections between the two.

Given a polynomial whose roots we wish to determine, we associate a field and a group to the polynomial and studying the properties of that group we discover what are the possibilities of finding the roots of the polynomial by elementary algebraic means.

Aims:

To study the unsuspected connections between polynomial equations and properties of the automorphism groups of field extensions.

To see how the same methods can be employed to decide whether certain geometric constructions (like the squaring of a circle or the duplication of a cube) can be carried out using only a straight edge and a compass.

Assessable learning outcomes:

Find the splitting fields of certain polynomials and their Galois Groups; decide whether the polynomial is solvable by radicals.

Examine whether certain geometric constructions are possible by straight edge and compass.

Additional outcomes:

Outline content:

Finite extensions and algebraic extensions of fields. The splitting field of a polynomial and its Galois Group over a given field.

The Galois correspondence and the fundamental theorem of Galois Theory. Solvable groups and the solvability by radicals of a polynomial.

Geometric constructions by straight edge and compass.

Brief description of teaching and learning methods:

Note that the 30-hours listed against 'Other' in the Independent Study Hours table is for problem solving.