Accessibility statement

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’ll develop your understanding of group theory and real analysis and see how to apply some of these ideas to cryptography and fractals.

This module is based around six books, each 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 identify the 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 allow you to gain the higher grades.

Number Theory

In the first book, we study integers and prime numbers. In particular, we look at classical problems that require integer solutions, such as finding all integer solutions to the equation 2x + 11y = 5. We also develop methods for solving linear congruences such as ax² ≡ 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 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 us to generalise 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 resolves 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 examine the implications of our new definition of distance for understanding what it means for something to be connected. The book culminates in an introduction to the theory of fractals.

The module website includes a non-assessed reader 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.

The full content list is on the Open mathematics and statistics website.

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.

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

They’ll help by:

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

Course work includes:

Each module book has an associated online quiz. You can attempt these quizzes as many times as you wish; they don't count towards your final grade.

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

We regularly review the assessments in our modules, so we may update the examination method used for this module to an in-person exam or an online invigilated exam. When we are making such a change, we will make it clear on this page. If we need to make a change after you have registered, we will notify you as soon as possible. If you have additional requirements, we will support you to complete your assessments.

You’ll have access to a module website, which includes:

Additionally, the website includes:

We also provide physical:

The OU strives to make all aspects of study accessible to everyone, and this Accessibility Statement outlines what studying M303 involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

To find out more about what kind of support and adjustments might be available, contact us or visit our Disability support website.

Further pure mathematics (M303) starts once a year – in October.

It will next start in October 2027.

We expect it to start for the last time in October 2029.

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