# MMath&Phys Mathematics and Physics

Year of entry: 2023

## Course unit details:Galois Theory

Unit code MATH42121 15 Level 4 Semester 1 No

### Overview

Galois theory is one of the most spectacular mathematical theories. It establishes a beautiful connection between the theory of polynomial equations and group theory. In fact, many fundamental notions of group theory originate in the work of Galois. For example, why are some groups called 'soluble'? Because they correspond to the equations which can be solved! (Solving here means there is a formula involving algebraic operations and extracting roots of various degrees that expresses the roots of the polynomial in terms of the coefficients.) Galois theory explains why we can solve quadratic, cubic and quartic equations, but no formulae exist for equations of degree greater than 4. In modern language, Galois theory deals with 'field extensions', and the central topic is the 'Galois correspondence' between extensions and groups. Galois theory is a role model for mathematical theories dealing with 'solubility' of a wide range of problems.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Algebraic Structures 2 MATH20212 Pre-Requisite Compulsory
Algebraic Structures 1 MATH20201 Pre-Requisite Compulsory
Group Theory MATH32001 Pre-Requisite Compulsory

Students must have taken MATH20212 and (MATH32001 OR MATH42001).

It is recommended that Year 3 students consult with the lecturer before signing up for this course.

Students are not permitted to take, for credit, MATH42122 in an undergraduate programme and then MATH62122 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

To introduce students to a sophisticated mathematical subject where elements of different branches of mathematics are brought together for the purpose of solving an important classical problem.

### Learning outcomes

On successful completion of this course unit students will be able to:

• Provide the definition of splitting field, finite field extension, algebraic extension, Kummer extension, normal extension, Galois group of an extension or polynomial, solvable group and solvability of a polynomial by radicals.
• Prove basic properties of finite fields such as: their cardinality is a power of a prime, the multiplicative group is cyclic, the Frobenius map is an automorphism.
• State the fundamental theorem of Galois theory and the Galois correspondence; as well as, Galois theorem on the radical-solvability of polynomial.
• State, prove, and use Eisensteins criterion for the irreducibility of polynomials over the rationals; as well as, compute the degree and find a basis of the splitting field of a polynomial of low degree.
• Identify basic group-theoretic properties (such as its order or being cyclic, abelian, and/or solvable) of the Galois group of a finite field extension of low degree.
• Compute the Galois group of a field extension of low degree; as well as, compute the fixed fields of it subgroups and identify those that are normal extensions.
• Explain how one can use Galois theory to prove that polynomials of degree less than five are solvable by radicals, while the general quintic equation is not.

### Syllabus

1. Introduction and preliminaries: fields, vector spaces, homogeneous linear systems, polynomials. [4 lectures]
2. Field extensions, algebraic elements, Kronecker's construction. 
3. Splitting fields. 
4. Group characters, automorphisms and fixed fields. 
5. Normal extensions, separable polynomials, formal derivatives. 
6. The Fundamental Theorem of Galois Theory, Galois groups of polynomials, examples of the Galois correspondence. 
7. Finite fields, roots of unity. 
8. Kummer extensions, 
9. Solutions of polynomial equations by radicals and an insolvable quintic. 

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: one in-class test, weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

The lecture notes are self-contained, however the following texts are recommended:

E. Artin, Galois Theory, Dover Publications 1998. This provides a concise account of the material covered in the course.
I Stewart, Galois Theory, 2nd edition, Chapman and Hall. Contains the key ideas, together with some historical comment, illustrations, and exercises with selected solutions.
J B Fraleigh, A First Course in Abstract Algebra, 5th edition, Addison-Wesley 1967. Chapters 8 and 9 of this book are directly relevant to the course, and contain a number of exercises with solutions. Earlier chapters may be useful for revision of pre-requisite material.

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 126

### Teaching staff

Staff member Role
Peter Rowley Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.