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# Galois Theory (MSc) - MAT00074H

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• Department: Mathematics
• Module co-ordinator: Dr. Emilie Dufresne
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2020-21
• See module specification for other years: 2019-20

## Module summary

This is for postgraduate students only.

## Related modules

• None

### Prohibited combinations

• None

MSc students should have taken a suitable first course in Pure Mathematics

## Module will run

Occurrence Teaching cycle
A Spring Term 2020-21

## Module aims

• To introduce one of the high points of 19th century algebra.

• To exhibit the unity of mathematics by using ideas from different modules.

• To show how very abstract ideas can be used to derive concrete results.

## Module learning outcomes

• Construct fields as quotients of polynomial rings by maximal ideals.

• Irreducibility criteria.

• Any irreducible polynomial has a suitable extension field.

• The degree of a field extension, and its multiplicativity.

• The form of the elements in a simple extension.

• Splitting fields.

• The allowable operations with straightedge and compasses.

• Constructible complex numbers form a field which is closed under extraction of square roots.

• Each constructible number lies in a field of degree a power of two over Q, and the consequences of that fact.

• Some of the results about automorphisms of an extension field, and the fixed field of a group of automorphisms.

• The Galois Correspondence Theorem, and the ability to apply it to straightforward examples.

• Some of the consequences of the Galois Correspondence.

## Assessment

Task Length % of module mark
University - closed examination
Galois Theory
2 hours 100

Pass/fail

### Reassessment

Task Length % of module mark
University - closed examination
Galois Theory
2 hours 100

## Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.