Undergraduate Course: Group and Galois Theory (MATH10078)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Course type | Standard |
Availability | Available to all students |
| Credit level (Normal year taken) | SCQF Level 10 (Year 4 Undergraduate) |
Credits | 20 |
| Home subject area | Mathematics |
Other subject area | None |
| Course website |
None |
Taught in Gaelic? | No |
| Course description | 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. |
Information for Visiting Students
| Pre-requisites | None |
| Displayed in Visiting Students Prospectus? | Yes |
Course Delivery Information
|
| Delivery period: 2014/15 Full Year, Available to all students (SV1)
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Learn enabled: Yes |
Quota: None |
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Web Timetable |
Web Timetable |
| Course Start Date |
15/09/2014 |
| Breakdown of Learning and Teaching activities (Further Info) |
Total Hours:
200
(
Lecture Hours 44,
Seminar/Tutorial Hours 10,
Summative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
139 )
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| Additional Notes |
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| Breakdown of Assessment Methods (Further Info) |
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %
|
| No Exam Information |
Summary of Intended 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. |
Assessment Information
| Coursework 20%, Examination 80% |
Special Arrangements
| None |
Additional Information
| Academic description |
Not entered |
| Syllabus |
· 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 |
| Transferable skills |
Not entered |
| 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 )
- J J Rotman, Galois Theory
- I Stewart, Galois Theory (QA214 Ste)
- D J H Garling, A Course in Galois Theory (QA211 Gar)
- J-P Escofier, Galois Theory (QA174.2 Esc)
- J-P Tignol, Galois¿ theory of algebraic equations (QA211 Tig)
- H M Edwards, Galois Theory (QA274 Edw) |
| Study Abroad |
Not entered |
| Study Pattern |
Not entered |
| Keywords | GGT |
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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