Groups, Rings and Fields - 2023 entry

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).
 

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.

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.
 

- 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.
 

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

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

 

Reading list for this module: