Unit information: Galois Theory in 2020/21

Unit name	Galois Theory
Unit code	MATHM2700
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Walling
Open unit status	Not open
Pre-requisites	MATH21800 Algebra 2 and MATH33300 Group Theory
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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.

Unit Description

Consider a field, such as the rational numbers, and consider a larger field containing it, which can also be thought of as a vector space over the original field. One question that can be asked is: what symmetries (or automorphisms) of the bigger field exist that act as the identity on the smaller field? The Galois Correspondence connects the answer to this question with the properties of a group of permutations of the roots of a polynomial. This relationship can be used in different directions to translate a problem from one part of algebra to another part where it may be easier to solve.

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.

Intended Learning Outcomes

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.

Transferable Skills

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

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References

Recommended

• Emil Artin, Galois Theory, New ed. Dover, 1998
• D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, 1986
• Ian Stewart, Galois Theory, 3rd ed. Chapman & Hall, 2003