• Inactive for Year: 2024/25

• Module Leader(s): Dr James Waldron

• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus

Semesters

Your programme is made up of credits, the total differs on programme to programme.

Aims

To present an introduction to Galois theory, bringing together ideas from several other modules (linear algebra, algebra, group theory). To develop the theory of field extensions and their groups of automorphisms, culminating in the Galois correspondence. To apply the theory to find solutions of polynomial equations by radicals and to explain why and when these exist. If time, to consider applications to ruler-and-compass constructibility of geometric figures.

Module Summary

Galois theory originates from the work of Evariste Galois, on the existence of formulas for the solutions of polynomial equations. The standard formula for the roots of a quadratic equation can be generalised to give a formula for roots of a polynomial of degree 3 or of degree 4. However, as Abel showed at the beginning of the 19th century, no such formula can exist for polynomials of degree 5 or more. To explain this, Galois’ idea was to consider certain symmetries between the solutions of a polynomial equation. Nowadays, these symmetries are described in terms of group theory, a subject which can be viewed as beginning in the work of Galois. In modern terminology, Galois demonstrated how solutions of a polynomial equation correspond to certain groups of symmetries of a field extension. Moreover, properties of solutions of an equation may be read off from properties of the group and vice-versa. The module covers the field and group theory necessary to establish this theory and applies the theory to study the existence and computation of solutions of polynomials. If time allows, the theory will also be applied to the construction of regular polygons, using ruler and compasses.

Outline Of Syllabus

Fields and polynomials: field extensions and their degrees; algebraic and transcendental extensions. Soluble groups: derived series, simple groups. Galois group of a field extension: Fundamental Theorem of Galois Theory. Solutions of equations by radicals. If time, ruler and compass constructions: squaring the circle, regular polygons.

Teaching Methods

Teaching Activities

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams

Other Assessment

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Exam problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Reading Lists

• MAS8752's Reading List

Timetable

• Timetable Website: www.ncl.ac.uk/timetable/
• MAS8752's Timetable