Groups, Actions & Galois Theory - MAT00099H
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 2 2025-26 |
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:
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Apply group actions to numerous topics in group theory.
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Construct Sylow subgroups and apply the Sylow theorems.
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Construct the various kinds of field extensions.
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Use the Galois correspondence to analyse intermediate fields.
Module content
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Group actions, orbits, stabilisers and the orbit-stabiliser theorem.
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Cauchy’s theorem, counting orbits and Burnside’s theorem.
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Homomorphism theorems
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Sylow theorems.
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Conjugacy.
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Fields as quotients of polynomial rings.
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Field extensions, splitting fields.
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Constructible numbers.
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The Galois group of an extension.
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The Galois correspondence and some applications.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 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.