Undergraduate Course: Group and Galois Theory (MATH10078)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 10 (Year 4 Undergraduate) |
Availability | Available to all students |
| SCQF Credits | 20 |
ECTS Credits | 10 |
| Summary | This is a course in abstract algebra, although connections with other fields will be stressed as often as possible. It begins with a systematic study of the basic structure of groups, finite and infinite. There will also be some ring theory. Then it will cover some of the jewels in the crown of undergraduate mathematics, drawing together groups, rings and fields to solve problems that resisted the
efforts of mathematicians for many centuries. The powerful central ideas of this course are now crucial to many modern problems in algebra, differential equations, geometry, number theory and topology. |
| Course description |
- Group actions & Sylow Theorems
- Homomorphisms, isomorphisms & factor groups
- Group Presentations
- Simple groups & Composition Series
- Polynomial rings & finite fields
- Classification of finite abelian groups
- PIDs & their modules
- Fields: examples, constructions and extensions
- Separability, normality & splitting fields
- Field automorphisms & Galois groups
- The fundamental theorem of Galois Theory
- Solvable groups and the insolubility of the general quintic
- Ruler and Compass constructions
- Calculation of Galois groups
- Transcendence
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Information for Visiting Students
| Pre-requisites | None |
| High Demand Course? |
Yes |
Course Delivery Information
| Not being delivered |
Learning Outcomes
1. Facility with the Sylow theorems, group homomorphisms and presentations, and the application of these in order to describe aspects of the intrinsic structure of groups, both abstractly and in specific examples.
2. Ability to manipulate composition series, through both the proof of abstract structural properties and the calculation of explicit examples.
3. Capacity to work with the classes of rings and fields appearing in the course, particularly specific calculations around finite fields and polynomials.
4. Facility with fields and their extensions, including expertise in explicit calculations with and constructions of examples with various relevant desired properties.
5. Ability to handle Galois groups, abstractly and in explicit examples, by using a variety of techniques including the Fundamental Theorem of Galois Theory and presentations of fields.
6. Capacity to explain and work with the consequences of Galois Theory in general questions of mathematics addressed in the course, such as insolubility of certain classes of equations or impossibility of certain geometric constructions.
7. Be able to produce examples and counterexamples illustrating the
mathematical concepts presented in the course.
8. Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.
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Reading List
Recommended :
- T S Blyth and E S Robertson, Groups (QA171.Bly )
- J F Humphreys, A Course in Group Theory (QA177 Hum)
- M A Armstrong, Groups and Symmetry (QA171 Arm )
- J J Rotman, The theory of groups: An introduction (QA171 Rot )
- J J Rotman, An introduction to the Theory of Groups (QA174.2 Rot )
Students are expected to have a personal copy of 'Galois Theory', Fourth Edition (Chapman and Hall / CRC) by Ian Nicholas Stewart. ISBN-13: 978-1482245820. |
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | GGTh |
Contacts
| Course organiser | Dr Susan Sierra
Tel: (0131 6)50 5070
Email: S.Sierra@ed.ac.uk |
Course secretary | Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk |
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