Mathematics

ECM2711 - Groups, Rings and Fields (2015)

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MODULE TITLEGroups, Rings and Fields CREDIT VALUE15
MODULE CODEECM2711 MODULE CONVENERDr Henri Johnston (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
Number of Students Taking Module (anticipated) 79
DESCRIPTION - summary of the module content

Building on the foundations established in ECM1706 or equivalent, you will learn about the key unifying structures that form a basis for any further modules in Algebra, Number Theory and Geometry at Stages 3 and 4

This module gives you the opportunity to further refine your skills in problem-solving, axiomatic reasoning and the formulation of mathematical proofs.

Students that take this course are expected to have a strong interest in Algebra. The level of difficulty of this course is considerably higher than that of Numbers, Symmetries and Groups (ECM1706).


Prerequisite modules: ECM1706 and  ECM1701 or equivalent

AIMS - intentions of the module

The aim of this module is to give you an introduction to the theory of groups, rings and fields, which are essential to further studies in algebra and related topics at Levels 3 and M.  The approach will be to study these structures both from an axiomatic basis, but also as a unifying tool for important examples from diverse topics in mathematics.  As such, you will need to understand both the abstract theory and its application to a range of examples.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

 

Module Specific Skills and Knowledge:
1 recall and apply key definitions in the theory of groups, rings and fields;
2 state, prove and apply core theorems in the theory of groups, rings and fields.
Discipline Specific Skills and Knowledge:
3 perform computations accurately;
4 use abstract reasoning to solve a range of problems.
Personal and Key Transferable / Employment Skills and Knowledge:
5 communicate your findings effectively in writing;
6 work independently and manage your time and resources effectively.

 

SYLLABUS PLAN - summary of the structure and academic content of the module

- 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;

- Orbit-Stabiliser 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, p-th 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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 44.00 Guided Independent Study 106.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 33 Lectures including example classes
Scheduled learning and teaching activities 11 Tutorials
Guided independent study 106 Lecture and assessment preparation; wider reading

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Exercises One sheet fortnightly All Verbal and generic feedback in tutorials
       
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Exam mark – verbal feedback on request
Coursework – based on questions submitted for formative assessment 20 30 hours in total All Annotated script and written feedback
         
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All above Written exam (100%) All August Ref/Def period
       
       

 

RE-ASSESSMENT NOTES

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.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: College to provide hyperlink to appropriate pages

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Wallace D.A.R. Groups Rings and Fields Springer 2001 000-3-540-76177-2 [Library]
Set Durbin, J. Modern Algebra: An Introduction Sixth John Wiley & Sons 2009 978-0-470-53035-1 [Library]
Set Cameron, P.J. Fields Introduction to Algebra Second Oxford Science Publications 2008 978-0-19-852793-0 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES ECM1706, ECM1701
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 5 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Friday 09 January 2015 LAST REVISION DATE Friday 09 January 2015
KEY WORDS SEARCH Algebra; rings; fields; groups.