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MTH2002  Algebra (2018)
MODULE TITLE  Algebra  CREDIT VALUE  30 

MODULE CODE  MTH2002  MODULE CONVENER  Dr Gihan Marasingha (Coordinator) 
DURATION: TERM  1  2  3 

DURATION: WEEKS  11  11  0 
Number of Students Taking Module (anticipated)  139 

In this module, you will explore some of the key techniques of modern algebra, including the theory of abstract vector spaces and ring theory. Whilst the roots of these topics lie in the arithmetic and geometry studied by the earliest mathematicians, much of the unifying power of this subject lies in the modern axiomatic treatment of this material dating from the 19th and 20th century. This material is essential to the study of many of our pure mathematics modules at levels 3 and M.
Prerequisite module: MTH1001.
This module aims to develop the theories and techniques of modern algebra, particularly in relation to rings, vector spaces and inner product spaces over arbitrary fields.
On successful completion of this module, you should be able to:
Module Specific Skills and Knowledge:
1 recall key definitions of rings and fields, and the most important examples of each;
2 understand the relationship between linear maps and matrices, and how the properties of each influence the solvability of systems of linear equations;
3 comprehend algorithms for solving linear equations and finding eigenvalues and eigenvectors in rigorous and formal terms.
Discipline Specific Skills and Knowledge:
4 tackle problems in many branches of mathematics that are linearisable, using the core skills of solving linear systems;
5 reveal sufficient knowledge of the fundamental algebraic concepts needed for subsequent studies in pure mathematics.
Personal and Key Transferable / Employment Skills and Knowledge:
6 appreciate that concrete problems often require abstract theories for their solution;
7 show the ability to monitor your own progress, to manage time, and to formulate and solve complex problems.
 rings and fields: review of field properties of Q, R and C; definitions of rings and fields; the fields Fp; other examples of rings, including Z and K[X] (for K a field); existence of greatest common divisors in Z and K[X]; extended Euclidean Algorithm;
 recap vector spaces, linear maps and key definitions and theorems; redevelopment of key results over arbitrary fields, where appropriate.
 characteristic and minimal polynomials; Jordan Canonical Form;
 normed and inner product spaces: bilinear forms and inner products; norms; unitary matrices; normal matrices and diagonalisability; dual spaces and examples; adjoint maps.
 review of group axioms and basic examples: cyclic, symmetric and dihedral groups;
 homomorphisms, kernel, image, isomorphisms;
 left and right cosets, normal subgroups;
 quotient groups, the first isomorphism theorem;
 group actions and permutation representations;
 group acting on itself by left multiplication;
 OrbitStabiliser Theorem, Orbit Counting Lemma;
 group acting on itself by conjugation, conjugacy classes, centre of a group, conjugacy in Sn, simple groups, A5 is simple;
 Sylow’s theorems;
 axioms for rings, examples: integers, integers modulo n, Matrix ring, polynomial ring (over C, R, and Q);
 units, zero divisors, integral domains, fields, field of fractions of an integral domain;
 rings homomorphisms, kernel, image, characteristic of a ring, pth power map in characteristic p;
 ideals: principal, prime, and maximal ideals;
 quotient rings, the first isomorphism theorem;
 polynomial rings over a field, and over an integral domain;
 principal ideal domain, unique factorisation domain; maybe: minimal polynomial;
 irreducibility criteria for polynomials: Gauss’s Lemma and Eisenstein’s criterion.
Scheduled Learning & Teaching Activities  76.00  Guided Independent Study  224.00  Placement / Study Abroad 

Category  Hours of study time  Description  
Scheduled learning and teaching activities 

Lectures including example classes  
Scheduled learning and teaching activities  10  Tutorials  
Guided independent study  224  Lecture and assessment preparation; wider reading 
Form of Assessment  Size of Assessment (e.g. duration/length)  ILOs Assessed  Feedback Method 

Exercise sheets  10 x 10 hours  All  Exercises discussed in tutorials: tutor feedback. 
Coursework  0  Written Exams  100  Practical Exams  0 

Form of Assessment  % of Credit  Size of Assessment (e.g. duration/length)  ILOs Assessed  Feedback Method 

Written exam – closed book (Jan)  30  2 hours  All  Via SRS 
Written exam – closed book (May)  70  2 hours  All  Via SRS 
Original Form of Assessment  Form of Reassessment  ILOs Reassessed  Time Scale for Reassessment 

All above  Written exam (100%)  All  August Ref/Def period 
If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.
If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.
information that you are expected to consult. Further guidance will be provided by the Module Convener
ELE: http://vle.exeter.ac.uk
Reading list for this module:
Type  Author  Title  Edition  Publisher  Year  ISBN  Search 

Set  Axler S, Gehring,F W, Ribet, K A  Linear Algebra done right  2nd  Springer  1997  9780387982588  [Library] 
Set  Cohn P.M.  Elements of Linear Algebra  1st  Chapman & Hall/CRC  1994  9780412552809  [Library] 
Set  Griffel D.H.  Linear algebra and its applications. Vol.1, A first course  Ellis Horwood Limited  1989  000074580571X  [Library]  
Set  Griffel D.H.  Linear algebra and its applications. Vol.2, More advanced  Ellis Horwood Limited  1989  000047021354X  [Library]  
Set  Wallace D.A.R.  Groups Rings and Fields  Springer  2001  0003540761772  [Library]  
Set  Durbin, J.  Modern Algebra: An Introduction  Sixth  John Wiley & Sons  2009  9780470530351  [Library] 
Set  Cameron, P.J.  Fields Introduction to Algebra  Second  Oxford Science Publications  2008  9780198527930  [Library] 
CREDIT VALUE  30  ECTS VALUE  15 

PREREQUISITE MODULES  MTH1001, MTH1002 

COREQUISITE MODULES 
NQF LEVEL (FHEQ)  5  AVAILABLE AS DISTANCE LEARNING  No 

ORIGIN DATE  Wednesday 11 January 2017  LAST REVISION DATE  Thursday 28 February 2019 
KEY WORDS SEARCH  Vector spaces; linear maps; scalar products; orthogonal vectors; linear independence; spanning sets; subspaces; Jordan form; adjoint; dual; rings; groups; fields; isomorphism; irreducibility; characteristic polynomial. 
