Algebra - 2019 entry

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.
 

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;

- 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.

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%.

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

Reading list for this module: