## 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: 2021/2

Module Convenor: Dr Basil Corbas

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.

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.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 4 Guided independent study: Exam revision/preparation 40 Advance preparation for classes 10 Other 26 Total hours by term 100 0 0 Total hours for module 100

Summative Assessment Methods:
 Method Percentage Written exam 100

Summative assessment- Examinations:

Two Hour examination

Summative assessment- Coursework and in-class tests:

Formative assessment methods:

Problem Sheets

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.
The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:

A mark of 40% overall.

Reassessment arrangements:

Two Hour Re-examination