Groups, Rings and Fields - MAT00046I
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
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 |
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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:
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apply general abstract concepts to particular groups, rings and fields
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understand and use the notion of proof in the context of the module
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apply the given theorems, such at the Fundamental Theorem of Homomorphisms for Groups, to both familiar and new examples
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demonstrate knowledge of direct and semidirect products
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understand and implement the division algorithm for rings of polynomials
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construct multiplication tables for finite fields
Module content
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Definition of group, examples and consequences of the axioms
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Symmetric groups
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Normal subgroups and conjugacy
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Group isomorphisms and homomorphisms
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Quotient groups and the homomorphism theorems
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External and internal direct and semidirect group products
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Classification theorems for groups
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Definition of rings, examples and consequences of the axioms
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Subrings and ideals
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Ring homomorphisms and isomorphisms; quotient rings
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Rings of polynomials with coefficients in any field
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Special rings: integral domains, principal ideal domains and fields
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Division in commutative rings with identity
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Prime and irreducible elements; unique factorization domains
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Prime and maximal ideals; their characterisation by quotients for commutative rings
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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
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M A Armstrong, Groups and Symmetry, Springer
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P M Cohn, Classic Algebra, Wiley-Blackwell
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J B Fraleigh, A First Course Abstract Algebra, Addison-Wesley
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C R and D A Jordan, Groups, Butterworth-Heinemann
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J F Humphreys, A Course in Group Theory, Oxford
- W Ledermann and A J Weir, Introduction to Group Theory, Longman