What you will study
This module is based around six books with each one developing a particular topic in pure mathematics.
The first book is concerned with the integers, and in particular with the solution of classical problems that require integer solutions. It begins by considering some elementary properties of the integers, such as divisibility and greatest common divisors. This leads to a method of solving the linear Diophantine equation ax + by = c, that is, finding solutions to the equation that are integers. In the second chapter, every integer greater than 1 is shown to be a unique product of primes, and results are obtained concerning the distribution of primes among the integers. In Chapter 3, methods are developed for solving linear congruences such as ax ≡ b (mod n) and in the final chapter the classical theorems of Fermat and Wilson are obtained.
The second book consolidates and builds on the group theory presented at OU level 2 of our curriculum in Pure mathematics (M208). You will learn enough about the structure of groups to completely determine, up to isomorphism, all groups of order less than 16. The introduction of direct products will also enable you to determine the structure of all finite Abelian groups. It ends with an introduction to the problem of classifying groups that are not given to be Abelian. On completion you should understand the structure of finite Abelian groups and be able to use the Sylow Theorems to analyse the structure of appropriate finite groups.
Numbers and Rings
In the first half of this book you will look at multiplicative functions and then return to congruences and consider the solution of quadratic congruences, ax2 ≡ b (mod n). This leads to Gauss’s law of quadratic reciprocity. The second half of this book contains an introduction to rings and the important idea of unique factorisation together with some applications to Number Theory, including techniques of solving some Diophantine equations.
Metric spaces I
This book introduces you to the theory of metric spaces: spaces in which there is a notion of distance between pairs of points. In the first chapter you see how the Euclidean notion of distance underlies the definition of continuity in the real line and the plane. In the second chapter, three key properties of this usual notion of distance are identified and used to define the idea of a metric. You will see how you can define metrics on spaces of functions and other abstract spaces. In Chapter 3 you learn how to construct and combine examples of metric spaces, including examples of distance defined for continuous functions, and in the last chapter you learn about open and closed sets in metric spaces.
Rings and Fields
This book consists of four chapters: Rings and Homomorphisms, Fields and Polynomials, Fields and Geometry, and Cryptography. The first chapter starts by introducing the construction of fields of fractions and then investigates rings derived from polynomials. It then looks at quotient rings and ideals, the ring theory analogues of quotient groups and normal subgroups and develops the concept of prime and maximal ideals. The Fields chapters look at many examples of fields, in particular finite fields, and a complete classification of finite fields is obtained. The third chapter includes investigations of ruler and compass constructions, resulting in the resolution of some famous problems of antiquity such as ‘squaring the circle’ or ‘trisecting the angle’. The chapter on cryptography includes some applications of finite fields.
Metric Spaces II
Finally, this book develops the theory of metric spaces by looking at the meaning of connectedness and understanding how theorems such as the extreme value theorem from real analysis can be extended to the context of metric spaces. This book culminates in an introduction to the theory of fractals where you can see how many common fractal sets can be viewed as fixed points of continuous maps on a very particular metric space.
There will be a reader on the module website that provides an overview of the historical development of topological and metric spaces, and modern algebra. Where appropriate the reader includes information and/or links about modern applications and unsolved/recently solved problems.
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.