MATH20310

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

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Table of Contents:

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.]