# Mathematics

## MTH2010 - Groups, Rings and Fields (2022)

MODULE TITLE CREDIT VALUE Groups, Rings and Fields 15 MTH2010 Prof Mohamed Saidi (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 120
DESCRIPTION - summary of the module content

In this module, you will explore some of the key techniques of modern algebra, including groups, rings, and fields. These topics have their roots in the desire to solve certain equations that arise from arithmetic and geometry.

The most familiar example of a ring is the set of all integers Z=...,-3,-2,-1,0,1,2,3... equipped with the usual operations of addition and multiplication. The familiar properties of these operations serve as a model for the axioms for rings. We can consider whether certain equations have solutions in rings such as the integers. For example, Fermat's Last Theorem famously asserts that if n is a fixed integer that is at least 3, then the equation x^n + y^n = z^n has no solutions for which x, y and z are non-zero integers. Though this problem is easy to state, its solution is extremely difficult: it was first stated in 1637 but the first complete and correct proof was given in 1994. Ring theory is essential for the fourth year module MTHM028 Algebraic Number Theory, which in turn lays the foundations for solving problems such as Fermat's Last Theorem.

Fields are special types of ring in which every non-zero element has a multiplicative inverse. Examples include the rational numbers Q, the real numbers R and the complex numbers C.

Group theory was introduced in the first year and will be developed further in this module. Not only does group theory underpin ring theory, but is also interesting and useful in its own right. For example, we shall see that group actions can be used to solve certain counting problems.

The material in this module is essential for the study of many of our pure mathematics modules at levels 3 and M, including MTH3004 Number Theory, MTH3026 Cryptography, MTH3038 Galois Theory, MTHM010 Representation Theory of Finite Groups, MTHM028 Algebraic Number Theory and MTHM029 Algebraic Curves.

Prerequisite module: MTH1001 (or equivalent).

AIMS - intentions of the module

This module aims to develop the theories and techniques of modern abstract algebra, particularly in relation to groups, rings, and fields. The main emphasis will be on rigorous definitions and proofs, but there will also be some applications such as how to solve certain counting problems.

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 key definitions concerning abstract algebraic structures of groups, rings and fields;

2 understand examples of each of these, significant results on their structure and the relation between them.

Discipline Specific Skills and Knowledge:
3 tackle problems in many branches of mathematics that require the use of groups, rings and fields;
4 reveal sufficient knowledge of the fundamental algebraic concepts needed for advanced studies in pure mathematics.

Personal and Key Transferable / Employment Skills and Knowledge:
5 appreciate that concrete problems often require abstract theories for their solution;
6 show the ability to monitor your own progress, to manage time, and to formulate and solve complex problems.

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

- review of group axioms and basic examples: cyclic, symmetric and dihedral groups;
- group 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);
- definition and examples of fields including Q, R and C;
- units, zero divisors, integral domains, fields, field of fractions of an integral domain;
- ring homomorphisms, kernel, image;
- quotient rings, the first isomorphism theorem;
- existence of greatest common divisors in Z and K[X];
- extended Euclidean Algorithm;
- polynomial rings over a field, and over an integral domain;
- ideals: principal, prime, and maximal ideals;
- prime and irreducible elements;
- principal ideal domain, unique factorisation domain.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 38 112
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 5 Tutorials Guided independent study 112 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 Exercise sheets 5 x 10 hours All Exercises discussed in tutorials: tutor feedback.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 10 90 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 90% 2 hours (January) All Written/verbal on request, SRS Coursework exercises 1 5% 15 hours All Annotated script and written/verbal feedback Coursework exercises 2 5% 15 hours All Annotated script and written/verbal 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 Written Exam* Written Exam (2 hours) (90%) All August Ref/Def period Coursework exercises 1 * Coursework exercises 1 (5%) All August Ref/Def period Coursework exercises 2 * Coursework exercises 2 (5%) All August Ref/Def period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

RE-ASSESSMENT NOTES
Deferrals: Reassessment will be by coursework and/or exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

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: http://vle.exeter.ac.uk

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]
Set Dummit, D S and Foote, R M Abstract Algebra Third Prentice Hall 2003 978-0471433347 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH1001
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Wednesday 26 February 2020 Thursday 04 August 2022
KEY WORDS SEARCH Rings; groups; fields; isomorphism; group actions; ideals