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Further pure mathematics

This module covers essential 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.

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Module code




  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.

Study level

Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU module levels correspond to these frameworks.
Level of Study
3 10 6

Study method

Module cost

Entry requirements

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M303 is a demanding module (even for a 60 point Level 3) and you do need to be sure you...
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An intensive and tough module. I followed the OU advice and left the module till the end of my studies....
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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.

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.

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.

Teaching and assessment

Support from your tutor

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

  • Marking your assignments (TMAs) and providing detailed feedback for you to improve.
  • Guiding you to additional learning resources.
  • Providing individual guidance, whether that’s for general study skills or specific module content.
  • Facilitating online discussions between your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part.


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.

Future availability

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

This page describes the module that will start in October 2024.

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


As a student of The Open University, you should be aware of the content of the academic regulations which are available on our Student Policies and Regulations website.

Course work includes:

6 Tutor-marked assignments (TMAs)
1 Interactive computer-marked assignment (iCMA)

Entry requirements

There is no formal pre-requisite study, but you must have the required mathematical skills.

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

Talk to an advisor if you’re not sure you’re ready.

Preparatory work

You should have some familiarity with the concepts covered in the Are you ready? quiz, and follow the advice in the quiz.

The key topics to revise include:

  • group theory
  • real analysis.

Pure mathematics (M208) is ideal preparation.


Start End England fee Register
05 Oct 2024 Jun 2025 £3636.00

Registration closes 05/09/24 (places subject to availability)

This module is expected to start for the last time in October 2029.

Additional Costs

Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

If your income is not more than £25,000 or you receive a qualifying benefit, you might be eligible for help with some of these costs after your module has started.

Ways to pay for this module

Open University Student Budget Account

The Open University Student Budget Accounts Ltd (OUSBA) offers a convenient 'pay as you go' option to pay your OU fees, which is a secure, quick and easy way to pay. Please note that The Open University works exclusively with OUSBA and is not able to offer you credit facilities from any other provider. All credit is subject to status and proof that you can afford the repayments.

You pay the OU through OUSBA in one of the following ways:

  • Register now, pay later – OUSBA pays your module fee direct to the OU. You then repay OUSBA interest-free and in full just before your module starts. 0% APR representative. This option could give you the extra time you may need to secure the funding to repay OUSBA.
  • Pay by instalments – OUSBA calculates your monthly fee and number of instalments based on the cost of the module you are studying. APR 5.1% representative.

Joint loan applications

If you feel you would be unable to obtain an OUSBA loan on your own due to credit history or affordability issues, OUSBA offers the option to apply for a joint loan application with a third party. For example, your husband, wife, partner, parent, sibling or friend. In such cases, OUSBA will be required to carry out additional affordability checks separately and/or collectively for both joint applicants who will be jointly and severally liable for loan repayments.

As additional affordability checks are required when processing joint loan applications, unfortunately, an instant decision cannot be given. On average the processing time for a joint loan application is five working days from receipt of the required documentation.

Read more about Open University Student Budget Accounts (OUSBA).

Employer sponsorship

Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.

More than one in ten OU students are sponsored by their employer, and over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees. 

  • Your employer just needs to complete a simple form to confirm how much they will be paying and we will invoice them.
  • You won’t need to get your employer to complete the form until after you’ve chosen your module.  

Credit/debit card

You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module. 

We accept American Express, Mastercard, Visa and Visa Electron. 

Mixed payments

We know that sometimes you may want to combine payment options. For example, you may wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Account (OUSBA).

Please note: your permanent address/domicile will affect your fee status and, therefore, the fees you are charged and any financial support available to you. The fee information provided here is valid for modules starting before 31 July 2025. Fees typically increase annually. For further information about the University's fee policy, visit our Fee Rules

This information was provided on 19/06/2024.

Can you study an Access module for free?

Depending on eligibility and availability of places, you could apply to study your Access module for free.

To qualify, you must:

  1. be resident in England
  2. have a household income of less than £25,000 (or be in receipt of a qualifying benefit)
  3. have not completed one year or more on any full-time undergraduate programme at FHEQ level 4 or above or successfully completed 30 credits or more of OU study within the last 10 years

How to apply to study an Access module for free

Once you've started the registration process, either online or over the phone, we'll contact you about your payment options. This will include instructions on how you can apply to study for free if you are eligible and funded places are still available.

If you're unsure if you meet the criteria to study for free, you can check with one of our friendly advisers on +44 (0)300 303 0069, or you can request a call back.

Not eligible to study for free?

Don't worry! We offer a choice of flexible ways to help spread the cost of your Access module. The most popular options include:

  • monthly payments through OUSBA
  • part-time tuition fee loan (you'll need to be registered on a qualification for this option)

To explore all the options available to you, visit Fees and Funding.

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

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

If you have a disability

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 pages.