Further pure mathematics

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. This is designed to help time-poor students to identify material necessary to understand later parts of the module. It is possible to follow the core route for some weeks and the standard route for others. Following the core route through the whole module should enable you to pass 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.

Groups
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.

You can find the full content list on the Open mathematics and statistics website.

Entry

There is no formal pre-requisite study, but you must have studied some university-level pure mathematics (as part of a module or by self-study).

You can check you're ready for M303 and see the topics it covers here.

Pure mathematics (M208) is ideal preparation.

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

• Primary device – A desktop or laptop computer. It's possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device.
• Peripheral device – Headphones/earphones with a built-in microphone for online tutorials.
• Our OU Study app operates on supported versions of Android and iOS.
• Operating systems – Windows 11 or latest supported macOS. Microsoft will no longer support Windows 10 as of 14 October 2025.
• Internet access – Broadband or mobile connection.
• Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable.

Teaching and assessment

Support from your tutor

You'll get help and support from an assigned tutor throughout your module.

They'll help by:

• marking your assignments and offering detailed feedback to help you improve
• providing individual guidance, whether that's for general study skills or specific module content
• guiding you to additional learning resources
• facilitating online discussions between your fellow students in the dedicated module and tutor group forums.

Online tutorials run throughout the module. Where possible, we'll make recordings available. While they're not compulsory, we strongly encourage you to participate.

Assessment

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

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.

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.