Groups, Rings and Fields - MAT00046I

«Back to module search

  • Department: Mathematics
  • Credit value: 20 credits
  • Credit level: I
  • Academic year of delivery: 2025-26

Module summary

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.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Pre-requisite modules:

  • Introduction to Pure Mathematics
  • Foundations and Calculus

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).

Module will run

Occurrence Teaching period
A Semester 2 2025-26

Module aims

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.

Module learning outcomes

By the end of the module, students will be able to:

  1. apply general abstract concepts to particular groups, rings and fields

  2. understand and use the notion of proof in the context of the module

  3. apply the given theorems, such at the Fundamental Theorem of Homomorphisms for Groups, to both familiar and new examples

  4. demonstrate knowledge of direct and semidirect products

  5. understand and implement the division algorithm for rings of polynomials

  6. construct multiplication tables for finite fields

Module content

  • 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

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Additional assessment information

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

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

  • 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

  • W Ledermann and A J Weir, Introduction to Group Theory, Longman