UNIVERSITY OF WAIKATO
Department of Mathematics (SCMS)
MATH310-10A Modern Algebra
Paper outline
Lecturers:
Dr Ian Hawthorn Rm: G3.03 Ph: 838-4466 Ext. 8217 Professor Kevin Broughan Rm: G3.22 Ph: 838-4423Lecturers’ consultation hours:
Dr Ian Hawthorn Thursday 2.00 – 3.00 pm Professor Kevin Broughan Thursday 3.00 – 5.00 pm
Paper schedule:
GROUP THEORY – Weeks 1-6 (Dr Ian Hawthorn)
RING THEORY AND FIELD THEORY – Weeks 7-12 (Prof. Kevin Broughan)
LECTURES: Monday 1.10 - 2.00pm G.3.33 Tuesday 4.10 - 5.00pm S.1.03 Thursday 12.00 - 12.50pm S.1.03 Friday 12.00 - 12.50pm K.G.11
TESTS:
There will be TWO TESTS as follows:
Friday 23 Apr 12.00 – 1pm K.G.11 Thursday 3 June 12.00 – 1pm S.1.03 TEST 1 will cover Dr Ian Hawthorn’s half and TEST 2 Prof. Kevin Broughan’s half.
Description of the paper:
This paper consists of two sections which will be taught in sequence. Dr Ian Hawthorn will teach the Group Theory section and Professor Kevin Broughan will teach the Ring Theory and Field theory section.
The FIRST half of paper will concentrate on Group Theory. Groups are the mathematical structures used to describe symmetry. They have wide application to other parts of mathematics and to mathematical physics.
Topics:
Definitions of Symmetry; Examples; Group axioms; Subgroups, cosets, Lagrange's theorem; Homomorphisms and isomorphisms, kernel and Image; cyclic groups; abelian groups; quotients and isomorphism theorems; normal subgroups, conjugacy, normalisers and centralisers, Permutation groups and group actions; Orbit-Stabiliser theorem;
Burnside counting; direct and semidirect products; fundamental theorem of abelian groups; Sylows Theorem and p-groups.
The SECOND half will cover rings and fields. A ring is an algebraic system with two operations behaving like addition and multiplication of numbers. A field is a commutative ring where each non- zero element has a multiplicative inverse.
Topics:
Definition of a ring and examples; subrings, ideals and factor rings; ring homomorphisms; field of quotients of an integral domain; Euclidean domains, principle ideal domains and unique factorization domains. Polynomial and matrix rings, factorization of polynomials, extension fields, algebraic extensions, finite fields, ruler and compass constructions.
Required textbook:
Contemporary Abstract Algebra (7th edn) by Joseph A. Gallian.
A copy of this text can be borrowed from the Mathematics Department office (G3.19).
Recommended reading:
Other books which cover the same material and which could be useful include Durbin’s book, Fraleigh’s book, and the book by Dummit and Foote.
Any book with a title similar to ‘Abstract Algebra’ or ‘Modern Algebra’ is likely to be relevant.
Please consult the lecturers if you have questions about the relevance of a book.
Assessment:
Internal assessment/examination ratio is the better of 0:1 or 1:1.
There will be TWO tests plus SEVERAL assignments, to be handed out in classes through the semester.
The internal mark will be made up of 25% from EACH test with the remainder based on the assignments, (25% from Dr Hawthorn’s assignments and 25% from Professor Broughan’s assignments).
Other information:
Your attention is drawn to the following policies and regulations contained in the 2010 University Calendar
Assessment Regulations 2005 (pg 119) Student Discipline Regulations 2008 (pg 697) Computer Systems Regulations 2005 (pg 715) Policy on the Use of Maori for Assessment (pg 126)
Ethical Conduct in Human Research and Related Activities Regs 2008 (pg105) Student Research Regulations 2008 (pg 103)
Student Complaints Policy (pg 712)
For other information please refer to website: http://www.math.waikato.ac.nz/studentinfo.html
Dr Ian Hawthorn
Professor Kevin Broughan 2010
