Further pure mathematics

This module introduces important topics in the theory of pure mathematics including: number theory; the algebraic theory of rings and fields; and metric spaces. You will develop your understanding of group theory and real analysis and will see how some of these ideas are applied to cryptography and fractals. To successfully study this module you must have a keen interest in developing your ability to write mathematical proofs and already have a sound knowledge of group theory, linear algebra, convergence of real sequences and series, and continuity of real functions, as provided by our level 2 module, Pure mathematics (M208).

Course facts
About this course:
Course code M303
Credits 60
OU Level 3
SCQF level 10
FHEQ level 6
Course work includes:
6 Tutor-marked assignments (TMAs)
1 Interactive computer-marked assignment (iCMA)
No residential school

What you will study

This module is based around six books with each one developing a particular topic in pure mathematics.

Number Theory

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 is an OU level 3 module. OU level 3 modules build on study skills and subject knowledge acquired from studies at OU levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with the OU.

This module is designed to follow on from Pure mathematics (M208). If you successfully study this OU level 2 module and understand all its content, you will have the necessary background and skills for this OU level 3 module.

To pass this module, you will need:

  • to spend an average of at least 16–18 hours per week engaging with the module material
  • to be comfortable reading and understanding mathematical arguments (such as the proofs in Pure mathematics (M208)) – the ability to apply and use results, without understanding the theory behind their use, is not likely to be sufficient
  • successful experience of writing correct mathematical arguments (such as showing that a fact is always true), as developed in Pure mathematics (M208)
  • a good knowledge and understanding of the following pure mathematics topics:
  • group theory including: normal subgroups; quotient groups; homomorphisms; and group actions
  • linear algebra including: finding determinants ; using elementary row operations; determining the kernel and image of linear maps; and the dimension theorem
  • real analysis including: proving the convergence of real sequences and series; proving the continuity of real functions; and the Mean Value Theorem.

If you have any doubts about your mathematical knowledge and experience, our diagnostic quiz will help determine whether you are ready for this module.

If you have any doubt about the suitability of the module, please speak to an adviser.

Preparatory work

There is no specific preparatory work required for this module but it may be helpful for you to revise group theory and continuity of real functions before the module begins.

If you have a disability or additional requirement

Written transcripts of any audio components and Adobe Portable Document Format (PDF) versions of printed material are available. Some Adobe PDF components may not be available or fully accessible using a screen reader and musical notation and mathematical, scientific, and foreign language materials may be particularly difficult to read in this way. Other alternative formats of the study materials may be available in the future.

Study materials

What's included

Six printed module books and a handbook (which can be taken into the examination).

A study planner, module reader, module forums, assessment materials, practice quizzes and optional supplementary information available via the module website.

You will need

A calculator would be useful for the number theory-related parts of the module, though it is not essential. A simple four-function (+ – x ÷) model would suffice.

Computing requirements

You will need a device with internet access to study this module as a web browser is used to access learning materials and activities. Any other computer-based activities you will need to carry out, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment, are specified in the module materials. If any additional software is needed for these tasks it will either be provided or is freely available. You may need administrative privileges to install software required for this module. Windows 10 S is not suitable as it restricts software installation to software available in the Windows Application Store.

Suitable devices are:

  • A Windows desktop or laptop running Windows 7 or later operating system
  • A Macintosh desktop or laptop running OS X 10.8 or later operating system.

Some software will not run on Linux, iOS or Android devices.

A netbook, tablet, smartphone or Linux computer that supports one of the browsers listed below may be suitable. However, these devices may not be suitable for some activities. If you intend to use one of these devices please ensure you have access to a suitable desktop or laptop device that uses the Windows or OS X operating system in case you are unable to carry out all activities on your mobile device.

Recent versions of the following browsers are suitable for carrying out web-based activities:

  • Safari
  • Chrome
  • Firefox
  • Edge

Or Internet Explorer 9 and above.

Using a browser upgraded to the latest version will maximise security when accessing the internet.

Using company or library computers may prevent you accessing some internet materials or installing additional software.

To be able to talk and listen in our online discussions you will need both a microphone and speakers/headphones.

Devices with small screens may make it difficult to view the material provided and carry out the activities. However, a device that has a resolution of at least 1024 pixels horizontally and also at least 768 pixels vertically should be adequate.

See our Skills for OU study website for further information about computing skills for study and educational deals for buying Microsoft Office software.

Teaching and assessment

Support from your tutor

You will have a tutor who will mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials (either face-to-face or online) or day schools that you are encouraged, but not obliged, to attend. Where your face-to-face tutorials are held will depend on the distribution of students taking the module.

Contact us if you want to know more about study with The Open University before you register.


The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.

Each of the six module books has an associated practice quiz on the module website. You can attempt these quizzes as many times as you wish and they do not count towards your final grade.

Each TMA is associated with a particular module book and consists of a mixture of questions: some of which contribute to your final grade, and some are developmental. The feedback you receive on your answers will help you to improve your knowledge and understanding of the study material and to develop important skills associated with the module.

Future availability

The details given here are for the module that starts in October 2017. It starts once a year – in October.

This course is expected to start for the last time in October 2021.

Professional recognition

This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.