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# Galois Theory - MAT00008H

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• Department: Mathematics
• Module co-ordinator: Dr. Haralampos Geranios
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2022-23

## Related modules

• None

### Additional information

Pre-requisite modules: students must have taken Pure Mathematics or Pure Mathematics Option 1.

## Module will run

Occurrence Teaching cycle
A Autumn Term 2022-23

## 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
Online Exam
Galois Theory
N/A 100

None

### Reassessment

Task Length % of module mark
Online Exam
Galois Theory
N/A 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.

## Indicative reading

I Stewart, Galois Theory, Chapman and Hall

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.