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). Here's what a student that started in October 2016 had to say:

If you enjoyed M208 pure mathematics, then I recommend you study M303. There are basically four topics of number theory, group theory, ring and field theory and metric spaces. There is also a little bit of applied mathematics to cryptography and ruler and compass constructions which is not examinable. The workload is what should be expected for a 60 credit level 3 module but the abstract nature of the subject matter can make it seem more at times. All the tutors and the module team provide outstanding support and for enthusiasts like myself there is extra-curricular material and some history of mathematics. I studied the third presentation of the module and I anticipate it will become even better as it matures.

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.

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

A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

A desktop or laptop computer with either:

  • Windows 7 or higher
  • macOS 10.7 or higher

The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

To participate in our online-discussion area you will need both a microphone and speakers/headphones.

Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

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.

If you have a disability

The OU strives to make all aspects of study accessible to everyone. The Accessibility Statement below outlines what studying this module involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

Mode of study

Printed materials are provided for the core module text. All of this module's study materials are also online; this includes PDFs of any printed materials, plus some items which are only provided online. Online-only materials include self-assessed quizzes. Online materials also include links to external resources, online forums and online tutorial rooms.

Tuition strategy

This module provides face-to-face and online tutorials. Each face-to-face tutorial has an online alternative. Although not compulsory, attendance at tutorials will help you consolidate your learning.

Mathematical and scientific expressions and notations

Mathematical symbols and notation are used extensively throughout the module and you will be required to use such notation within assessment.

Diagrams and other visual content

The study materials contain a considerable number of diagrams. Figure descriptions are provided for most figures.

Specialist reading material

In this module you will be working with specialist reading material which includes extensive mathematical notation. This is delivered both online and in printed form.


This module has Tutor-Marked Assignments (TMAs), which can be submitted online via the OU electronic TMA system or by post, and an Interactive Computer-Marked Assignment (iCMA) completed online. The End-of-Module Exam must be taken at an exam centre.


You will receive feedback from your tutor on your submitted Tutor-Marked Assignments (TMAs). This will help you to reflect on your TMA performance. You should refer to it to help you prepare for your next assignment. Solutions and some explanations will also be provided for the interactive Computer-Marked Assignment (iCMA).


All University modules are structured according to a set timetable and you will need time-management skills to keep your studies on track. You will be supported in developing these skills.

Future availability

Further pure mathematics starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2021.

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.