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Galois Theory - MAT00008H

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• Department: Mathematics
• Module co-ordinator: Dr. Emilie Dufresne
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2021-22
  • See module specification for other years: 2022-23

Related modules

Additional information

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

Module will run

Occurrence	Teaching period
A	Spring Term 2021-22

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 -less than 24hrs (Centrally scheduled) Galois Theory	2 hours	100

Special assessment rules

None

Reassessment

Task	Length	% of module mark
Online Exam -less than 24hrs (Centrally scheduled) 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.

Indicative reading

I Stewart, Galois Theory, Chapman and Hall