MA3003: Groups, Rings and Fields
| School | Cardiff School of Mathematics |
| Department Code | MATHS |
| Module Code | MA3003 |
| External Subject Code | G100 |
| Number of Credits | 10 |
| Level | L6 |
| Language of Delivery | English |
| Module Leader | Professor Roger Behrend |
| Semester | Autumn Semester |
| Academic Year | 2014/5 |
Outline Description of Module
First, groups, rings and fields are introduced. The basic theory of subgroups of a given group and the construction of factor groups is introduced, and then similar constructions are introduced for rings. As an application of these ideas, the ring of polynomials over a given field is introduced, and it is explained how to obtain and characterize new fields from this ring by taking appropriate quotients. The heart of the course is the theory of field extensions. Here ideas from linear algebra and abstract algebra are applied to characterise simple extensions.
Recommended Modules: MA0212 Linear Algebra
On completion of the module a student should be able to
- determine a set is a group / ring with respect to given binary operations
- check a given group / ring for certain special conditions
- show that a subset of a group / ring is a subgroup / subring or factor group / an ideal
- apply reducibility criteria to polynomials
- decide if a mapping between groups / rings is a homomorphism and determine the associated subgroups / subrings.
- perform algebraic calculations in field extensions
How the module will be delivered
27 – 50 minute lectures
Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.
Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.
Skills that will be practised and developed
Skills:
Application of general group properties to specific cases.
Recognition of shared features of different groups.
Testing groups, rings and fields for specific properties.
Algebraic calculation in extension fields.
Transferable Skills:
An appreciation of the possible need for advanced concepts to be applied in resolving elementary problems.
Recognition of the power of abstraction.
The ability to take an open minded approach when dealing with a novel situation.
How the module will be assessed
Formative assessment is carried out by means of regular tutorial exercises. Feedback to students on their solutions and their progress towards learning outcomes is provided during tutorial classes.
The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.
The examination paper has a choice of three from four equally weighted questions.
Assessment Breakdown
| Type | % | Title | Duration(hrs) |
|---|---|---|---|
| Exam - Autumn Semester | 100 | Groups, Rings, And Fields | 2 |
Syllabus content
- Groups, Rings, fields, polynomial rings, factor-rings.
- Generating sets.
- Cosets and Lagrange's Theorem.
- Homomorphisms, normal groups and factor groups.
- The first isomorphism theorem.
- Structure of cyclic groups.
- Fields, characteristics, finite fields, extension fields.
- Algebraic and transcendental elements.
- Degrees of extension fields.
Essential Reading and Resource List
Abstract Algebra, Dummit, D. S., & Foote, R. M., Prentice Hall
Background Reading and Resource List
Ring, Fields and Groups, Allenby, R. B. J. T., Arnold
Introduction to Group Theory, Ledermann, W., & Weir, A. J., Pearson.
Algebra, Lang, S., Addison-Wesley
An Introduction to Galois Theory, Baker, A, www.maths.gla.ac.uk/~ajb/dvi-ps/Galois.pdf