- Department: Mathematics
- Credit value: 20 credits
- Credit level: I
- Academic year of delivery: 2024-25
- See module specification for other years: 2023-24
Groups, rings and fields are abstract algebraic structures. Groups measure symmetry; rings occur naturally as rings of matrices or polynomials; fields are the very special rings associated with vector spaces.
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Pre-requisite modules:
This module is the second part of the Pure Mathematics stream, and as such must be taken with Metric Spaces (even though the content of that module is not pre-req for this module).
Occurrence | Teaching period |
---|---|
A | Semester 2 2024-25 |
This module focuses on developing mathematical theories from axioms in the context of groups, rings and fields. These three themes share many common aspects, progressing in a rigorous manner with a focus on proof, though applications and connections with other areas of mathematics are never far from sight. By the end of the module, students will appreciate the scope and power of abstract algebra, and have a thorough grounding for further study in Stages 3 and 4.
By the end of the module, students will be able to:
apply general abstract concepts to particular groups, rings and fields
understand and use the notion of proof in the context of the module
apply the given theorems, such at the Fundamental Theorem of Homomorphisms for Groups, to both familiar and new examples
demonstrate knowledge of direct and semidirect products
understand and implement the division algorithm for rings of polynomials
construct multiplication tables for finite fields
Definition of group, examples and consequences of the axioms
Symmetric groups
Normal subgroups and conjugacy
Group isomorphisms and homomorphisms
Quotient groups and the homomorphism theorems
External and internal direct and semidirect group products
Classification theorems for groups
Definition of rings, examples and consequences of the axioms
Subrings and ideals
Ring homomorphisms and isomorphisms; quotient rings
Rings of polynomials with coefficients in any field
Special rings: integral domains, principal ideal domains and fields
Division in commutative rings with identity
Prime and irreducible elements; unique factorization domains
Prime and maximal ideals; their characterisation by quotients for commutative rings
Irreducible polynomials and quotient fields
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
There will be five formative assignments with marked work returned in the seminars. At least one of them will contain a longer written part, done in LaTeX
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy
M A Armstrong, Groups and Symmetry, Springer
P M Cohn, Classic Algebra, Wiley-Blackwell
J B Fraleigh, A First Course Abstract Algebra, Addison-Wesley
C R and D A Jordan, Groups, Butterworth-Heinemann
J F Humphreys, A Course in Group Theory, Oxford