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# 2016/17 Taught Postgraduate Module Catalogue

## MATH5246M Fields and Galois Theory

### 20 creditsClass Size: 25

Module manager: Dr Rudolf Tange
Email: R.H.Tange@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2016/17

### Pre-requisite qualifications

MATH2025 or MATH2026 equivalent.

This module is not approved as an Elective

### Module summary

Mathematicians regard this theory as one of the pinnacles of undergraduate mathematics - it brings together beautiful ideas from different parts of the undergraduate curriculum, particularly from Algebraic Structures - groups, rings and fields all interact in a nontrivial way. The module provides a good example of deeper level mathematics, hinting at the hidden depths of mathematics beyond undergraduate level, at research level etc. It is where everything comes together.In Stewart's book on Galois Theory, 2004, he writes: "Galois theory is a showpiece of mathematical unification, bringing together several different branches of the subject and creating a powerful machine for the study of problems of considerable historical and mathematical importance."

### Objectives

On completion of this module, students should be able to:

a) prove and use basic results about field extensions, including computing degrees and finding minimal polynomials.
b) understand and prove the basic connections between Galois groups and field extensions.
c) compute the Galois group of simple field extensions and field extensions arising from polynomials.
d) determine whether or not a given polynomial is solvable by radicals.

### Syllabus

- Field extensions, including the tower law, algebraic and transcendental elements, minimal polynomials, norm and trace.
- Field embeddings and Artin's Extension Theorem. Linear Independence of Characters.
- Splitting fields and normal extensions.
- Separable extensions and the Frobenius map.
- Galois extensions, and their characterisation as finite, separable and normal extensions. The Galois Correspondence and the - Primitive Element Theorem.
- Orbits under the Galois group, and their relationship to the norm and trace.
- Cyclotomic extensions, cyclic extensions and Hilbert's Theorem 90.
- Radical extensions and solvable extensions. In solubility of certain polynomials of degree five.
- Applications to symmetric functions and finite fields, and the explicit formulae for roots of polynomials of degrees three and four.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

157 hours:
- 87 hours studying the lecture notes
- 30 hours tackling the exercises
- 20 hours exam revision
- 20 hours tackling past papers.

### Opportunities for Formative Feedback

Regular exercise sheets and example classes.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

The reading list is available from the Library website

Last updated: 08/04/2016

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