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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).

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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.
3 10 6
Study method
Distance Learning
Module cost
See Module registration
Entry requirements
See Entry requirements

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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.

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.

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

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.

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.


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)
    No residential school

    Course satisfaction survey

    See the satisfaction survey results for this course.

    Entry requirements

    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.


    Start End Fee
    - - -

    No current presentation - see Future availability

    This module is expected to start for the last time in October 2021.

    Additional Costs

    Study costs

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

    If you're on a low income 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, Maestro (UK only), Mastercard, Visa/Delta 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).

    For more information about combining payment options, speak to an adviser or book a call back at a time convenient to you.

    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 fees and funding information provided here is valid for modules starting before 31 July 2019. Fees normally increase annually in line with inflation and the University's strategic approach to fees. 

    This information was provided on 17/01/2019.

    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. 

    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 website.