Introductory description

Galois theory is the study of solutions of polynomial equations. You know how to solve the quadratic equation ax2+bx+c=0 by completing the square, or by that formula involving plus or minus the square root of the discriminant b2−4ac. The cubic and quartic equations were solved “by radicals” in Renaissance Italy. In contrast, Ruffini, Abel and Galois discovered around 1800 that there is no such solution of the general quantic. Although the problem originates in explicit manipulations of polynomials, the modern treatment is in terms of felid extensions and groups of “symmetries” of fields. For example, a genral qunitic polynomial over Q has five roots α1.….α5, and the corresponding symmetry group is the permutation group S5 on these.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Developing the algebraic tools to understand the solutions of polynomial equations and further applications. In particular, we cover fields, field extensions, separable and normal extensions, automorphism groups of fields and their extensions, Galois extensions, the Fundamental Theorem of Galois theory, soluble groups and radical extensions, insoluble quintics, impossibility of trisecting an angle and squaring the circle.

Learning outcomes

By the end of the module, students should be able to:

• Solution by radicals of cubic equations and (briefly) of quartic equations.
• The characteristic of a field and its prime subfield. Field extensions as vector spaces.
• Factorisation and ideal theory in the polynomial ring k[x]; the structure of a simple field extension.
• The impossibility of trisecting an angle with straight-edge and compass.
• The existence and uniqueness of splitting fields.
• Groups of field automorphisms; the Galois group and the Galois correspondence.
• Radical field extensions; soluble groups and solubility by radicals of equations.
• The structure and construction of finite fields.

Indicative reading list

Books: DJH Garling, A course in Galois theory, CUP.
IN Stewart, Galois Theory, Chapman and Hall.

Subject specific skills

Fields and their automorphism groups are key objects for anyone continuing to study algebra, number theory, algebraic geometry etc. Students will also learn how to use a group of automorphisms to study an object, this is used throughout pure mathematics, applied mathematics, theoretical physics, and even biology and chemistry (in studying symmetries of molecules).

Transferable skills

Problem solving, logical reasoning and quantitative analytical skills are all tested and improved in this module. Moreover, the students are required to use knowledge from a various range of background modules and link them together to gain understanding of the topics in this module; a key skill used throughout every day and working life.