# MATH20310 Groups, Rings and Fields

This module is an introduction to group theory, ring theory and field theory.

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Chapter 1. Group Theory: First Examples and Definition; Permutations and the
Symmetric Group; Cycles and Disjoint Cycles; Transpositions, Even Permutations
and the Alternating Group; Cyclic Groups; More Examples of Finite Groups;
Subgroups; Cosets; Normal Subgroups; Quotient Groups; Homomorphisms and
Isomorphisms; Cayley’s Theorem; The First Isomorphism Theorem ; Automorphisms;
The Second and Third Isomorphism Theorems; The Correspondence Theorem; Finite
Simple Groups; Finite Abelian Groups.

Chapter 2. Ring Theory: Rings; Homomorphisms and Isomorphisms; Subrings;
Integral Domains; Units (a.k.a. Invertible Elements); Fields; Field of
Fractions of an Integral Domain; Image and Kernel of a Ring Homomorphism;
Ideals; Quotient Rings.

Chapter 3. Commutative Rings: Introduction; Polynomial Rings; The Division
Algorithm in F [X ] ; Finitely Generated Ideals; Principal Ideal Domains;
Associate, Irreducible and Prime Elements in Integral Domains; Greatest Common
Divisors; Euclidean Domains; Unique Factorization Domains; Prime Ideals;
Maximal Ideals.

Chapter 4. Field Theory: The Degree of a Field Extension; Prime Field and
Characteristic; Constructing New Fields from Old Fields; Roots and
Reducibility; Irreducibility in Q[X ]; Simple Field Extensions (Simple
Transcendental Extensions; Simple Algebraic Extensions); The Delian Problem.
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It is highly recommended that students have taken the module MATH10040 Numbers
and Functions prior to taking this module.

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[Disclaimer: module content and assessment strategies may be subject to minor
changes during the trimester. These changes may not be reflected in this module
descriptor at that time, but will be clearly communicated to all students via
other means.]

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this module the student should be able to explain what groups, rings, fields, their substructures, associated structures and associated maps are. The student should be able to give examples of these. The student should be able to explain the fundamental results concerning these structures, and solve routine as well as unseen problems.

Student Effort Hours:
Student Effort Type Hours
Lectures

30

Tutorial

6

Specified Learning Activities

30

Autonomous Student Learning

50

Total

116

Approaches to Teaching and Learning:
The intention is to have all contact hours face-to-face.

Some of the lectures may be in blended or on-line format.
Requirements, Exclusions and Recommendations
Learning Recommendations:

MATH 10270: Linear Algebra in the Mathematical Sciences

Module Requisites and Incompatibles
Incompatibles:
MST20010 - Algebraic Structures

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Class Test: In-term test. Unspecified n/a Standard conversion grade scale 40% No

20

No
Examination: End of trimester exam. 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

80

No

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour