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Groups, Actions & Galois Theory - MAT00099H

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• Department: Mathematics
• Module co-ordinator: Dr. Brent Everitt
• Credit value: 20 credits
• Credit level: H
• Academic year of delivery: 2023-24
  • See module specification for other years: 2024-25

Module summary

Groups measure the symmetry of concrete and abstract objects. This course continues their story from second year with an emphasis on the concept of a group action. The second half of the course links group theory with ring theory and shows how groups can be used to measure the symmetries of polynomial equations.

Related modules

Module will run

Occurrence	Teaching period
A	Semester 1 2023-24

Module aims

Groups measure the symmetry of concrete and abstract objects. This course continues their story from second year with an emphasis on the concept of a group action. The second half of the course links group theory with ring theory and shows how groups can be used to measure the symmetries of polynomial equations.

Module learning outcomes

By the end of the module, students will be able to:

1. Apply group actions to numerous topics in group theory.

2. Construct Sylow subgroups and apply the Sylow theorems to problems involving simple groups.

3. Construct the various kinds of field extensions.

4. Use the Galois correspondence to analyse intermediate fields.

Module content

• Group actions, orbits, stabilisers and the orbit-stabiliser theorem.

• Cauchy’s theorem, counting orbits and Burnside’s theorem.

• Homomorphism theorems

• Sylow theorems.

• Finite simple groups

• Conjugacy.

• Fields as quotients of polynomial rings.

• Field extensions, splitting fields.

• Constructible numbers.

• The Galois group of an extension.

• The Galois correspondence and some applications.

Assessment

Task	Length	% of module mark
Closed/in-person Exam (Centrally scheduled) Groups, Actions & Galois Theory	3 hours	100

Special assessment rules

None

Reassessment

Task	Length	% of module mark
Closed/in-person Exam (Centrally scheduled) Groups, Actions & Galois Theory	3 hours	100

Module feedback

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

Indicative reading

M A Armstrong, Groups and Symmetry, Springer UTM.

Ian Stewart, Galois Theory, Routledge.