# Advanced Algebra (M Level only)

Course category:
3rd year
Module code:
0590406
Year:
2011/12
Term:
Spring
Term:
Summer
Credits:
20
Lecturer:
Dr Michael Bate

## Aims

The aim of the module is to develop the theory of finite groups up to the Sylow Theorems and to study factorisation in commutative integral domains.

## Learning objectives

At the end of the module you should be able to...

• Work with group actions;
• Use Sylow's Theorems;
• Recognise examples of various types of integral domain;
• Use Euclid's algorithm in Euclidean domains.

## Syllabus

• Revision of basic group theory, finite abelian groups.  Group actions: orbits, stabilisers. Conjugation and the class equation; conjugacy in Sn and the simplicity of An. p-groups. Orbit counting theorem and examples. Finite direct products. Sylow Theorems with proofs and applications. Non-simplicity tests and techniques for showing that certain numbers cannot be the orders of simple groups.
• Revision of basic facts from ring theory: homomorphisms, maximal and prime ideals, integral domains and fields. Irreducible and prime elements. Different types of integral domain: Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains. Every Euclidean Domain is a Principal Ideal Domain. Every Principal Ideal Domain is a Unique Factorization Domain. Illustrative examples including the integers, Gaussian integers and polynomials over a field.

## Recommended texts

• J B Fraleigh, A First Course in Abstract Algebra, Addison-Wesley

## Teaching

• Spring: Weeks 2-10 Three hours per week
• Summer: Weeks 2-5 Four hours per week

Roughly two thirds of the contact time in this course is devoted to lectures. The other third consists of problem sessions, examples classes, revision, etc.

## Assessment

Three hour unseen examination towards the end of the summer term 90%

Coursework 10%. Note that coursework submitted after the advertised deadlines will be given a mark of zero.

## Elective information

This module is not available as an elective.