# Mathematics

## MTH2002 - Algebra (2019)

MODULE TITLE CREDIT VALUE Algebra 30 MTH2002 Dr Henri Johnston (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 11 0
 Number of Students Taking Module (anticipated) 203
DESCRIPTION - summary of the module content

In this module, you will explore some of the key techniques of modern algebra, including the theory of abstract vector spaces and ring theory. 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 extremely difficult: it was first stated in 1637 but the first complete and correct proof was given in 1994. Ring theory is essential for algebraic number theory, which in turn lays the foundations for solving problems such as Fermat's Last Theorem.

Abstract vector spaces are important objects in linear algebra, which has its origins in solving linear equations over a field such as the rational, real or complex numbers (fields themselves are special types of ring). The elements of a vector space can be somewhat abstract: for example, they can be certain types of function. However, it is precisely this abstraction that makes the theory of vector spaces such a powerful tool. They arise in almost every area of (pure and applied) mathematics and statistics, and so their importance is hard to overstate. For example, familiarity with these objects will deepen understanding of PDE (partial differential equation) methods.

Group theory was introduced in the first year and will be developed further in this module. Not only does group theory underpin both ring theory and the theory of abstract vector spaces, 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 and will also deepen understanding in many of our applied mathematics and statistics modules.

Prerequisite module: MTH1001.

AIMS - intentions of the module

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.

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 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.

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

- 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;

- 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 Guided Independent Study 76 224
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities
 66
Lectures including example classes
Scheduled learning and teaching activities 10 Tutorials
Guided independent study 224 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 10 x 10 hours All Exercises discussed in tutorials: tutor feedback.
Mid-term tests 2 x 1 hours All Feedback on marked sheets, class feedback

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Exam A – closed book (Jan) 50 2 hours All Via SRS
Written Exam B – closed book (May) 50 2 hours All Via SRS

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 A Written Exam A (50%, 2hr) All August Ref/Def period
Written Exam B Written Exam B (50%, 2hr) All August Ref/Def Period

RE-ASSESSMENT NOTES

In the case of module referral, the higher of the original assessment and the reassessment will be recorded for each component mark. In the case of module referral, the final mark for the module reassessment 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