Further pure mathematics

This module covers 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 an interest in pure mathematics and ideally should already have studied pure mathematics at level 2 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. A feature of the module is that we identify a core route through each book that is sufficient to pass the module but may not enable you to gain the higher grades.

Number Theory

In the first book we study the integers and prime numbers. In particular we look at classical problems that require integer solutions. For example finding all integer solutions to the equation 2x + 11y = 5. We also develop methods for solving linear congruences such as ax ≡ b (mod n) and in the final chapter we study the classical theorems of Fermat and Wilson.


In the second book we consolidate and build on the group theory presented at level 2 of our curriculum in Pure mathematics (M208). The book leads up to the classification of all finite abelian groups and ends with an introduction to the problem of classifying groups that are not abelian. On completion you will 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 we consider the solution of quadratic congruences, ax2 ≡ b (mod n). In the second half we use our knowledge of the integers to define and study the abstract algebraic structures known as rings.

Metric spaces I

In this book motivated by our understanding of how distance works for points in the plane, we define metrics, which can be used to give us an idea of distance between arbitrary objects (such as words or fractals). This allows to generalise the notion of what it means for a function to be continuous.

Rings and Fields

In this book we return to our study of algebra. We start by looking at polynomial rings and then continue our investigation of abstract algebraic structures. This unexpectedly leads to the resolution of some famous problems of antiquity such as 'squaring the circle' or 'trisecting the angle'. The final chapter shows how algebraic ideas underlie the modern theory of cryptography.

Metric Spaces II

In this book we return to metric spaces. We look at the implications of our new definition of distance for understanding what it means for something to be connected. This book culminates in an introduction to the theory of fractals.

There is a non-assessed 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.

Read the full content list here.


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).

To pass this module, you will need:

  • to spend an average of at least 16–18 hours per week engaging with the module material
  • some familiarity with the following pure mathematics topics: group theory; linear algebra and real analysis.

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). Informal online recorded lectures given by the module team. A study planner, history 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
  • Mac OS X 10.7 or higher

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

To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).

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 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 (M303) starts once a year – in October. This page describes the module that will start in October 2019. 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.