MSc Pure Mathematics

Fees for entry in 2027 have not yet been set. For reference, the fees for the academic year beginning September 2026 were as follows:

• MSc (full-time)
  UK students (per annum): £14,700
  International, including EU, students (per annum): £36,300
• MSc (part-time)
  UK students (per annum): £7,400
  International, including EU, students (per annum): £14,700

The fees quoted above are fully inclusive of tuition, administration and computational costs.

Fees for entry are subject to yearly review. The University reserves the right to increase your tuition fee by up to 7% each year for courses lasting more than one year, including to reflect rising costs associated with delivering our educational and wider student experience. Postgraduate fees information .

Always contact the admissions team if you are unsure which fee applies to your qualification award and method of attendance.

International student CAS deposit

Self-funded international applicants are required to pay a deposit of £2500 towards their tuition fees before a confirmation of acceptance for studies (CAS) is issued. Some applicants will be required to pay a higher deposit. More information on tuition fee deposits .

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

The Martingale Foundation helps provide access to postgraduate mathematics study for UK students facing financial barriers by offering fully-funded MSc and PhD programmes. Find more information on the Martingale Foundation website . Applications for 2027 scholarships open Wednesday 12 August 2026.

Our Manchester Master's Bursaries are aimed at widening access to master's courses by removing barriers to postgraduate education for students from underrepresented groups.

We also welcome the best and brightest international students each year and reward excellence with a number of merit-based scholarships. See our range of master’s scholarships for international students .

If you have completed or are in the final year of an undergraduate degree at The University of Manchester, you may be eligible for a discount of 10% on tuition fees for postgraduate study. Find out if you're eligible and how to apply .

For more information on master's tuition fees and studying costs, visit the University of Manchester funding for master's courses website.

Number theory is arguably one of the oldest and most fascinating branches of mathematics. This fascination stems from the fact that there are a great many theorems concerning the integers, which are extremely simple to state, but turn out to be rather hard to prove.
 

The fundamental objects in algebraic number theory are finite field extensions of Q; so-called number fields. To a number field k one associates a ring O_k  called its ring of integers. This ring behaves in some respects like the usual ring of integers Z, however many well know properties of Z do not pass over; the most important being that the fundamental theorem of arithmetic can fail in O_k .
 

The main focus of this course is on the failure of the unique factorisation. We also give a number applications to the study of certain diophantine equations.

Please note

Students are not permitted to take, for credit, MATH42132 in an undergraduate programme and then MATH62132 in a postgraduate programme at the University of Manchester, as the courses are identical

To show how tools from algebra can be used to solve problems in number theory.

On successful completion of this course unit students will be able to: 

• define the basic notions of algebraic number theory, such as algebraic numbers and integers, conjugates, number fields, rings of integers, norm, trace and discriminant, fractional ideals, class groups and lattices,
• describe the additive and multiplicative structure of a number field and it’s the ring of integers using the proper algebraic terminology,
• perform basic computations with algebraic integers in a simple number field, such as addition and multiplication, finding inverses and computing the minimal polynomial,
• identify the ring of integers and the discriminant of simple examples, such as quadratic and cyclotomic fields, and justify the identification,
• summarise a procedure to factorise prime numbers into prime ideals of a ring of integers and apply it in the case of simple number fields, such as quadratic fields,
• re-formulate statements concerning the existence of certain algebraic integers in terms of lattice points and apply Minkowski’s first theorem to prove them,
• compute class numbers and class groups of simple number fields, such as quadratic fields,
• solve simple Diophantine equations using factorisations of algebraic integers and ideals.

 

 

Syllabus

Fields and rings
- Review of required tools from the theory of fields and rings
- Field extensions, ideals, maximal ideals, prime ideals
- Euclidean domain  => PID  => UFD  => integral domain

Number fields
- Definitions and basic examples
- Embeddings into the real and complex numbers
- Field norms and trace

Rings of integers
- Integral closures
- Definitions and basic properties
- Discriminants
- Calculation for quadratic field extensions and cyclotomic fields

Unique factorisation of ideals 
- Prime ideals in rings of integers of number fields
- Unique factorisation into prime ideals

Geometry of numbers
- Lattices
- The Minkowski bound

Failure of unique factorisation
- Examples
- Definition and finiteness of the class group

Applications 
- Applications to non-linear Diophantine equations
- Some cases of Fermat’s last theorem
 

The teaching will be shared by Rose Wagstaffe and Raymond McCulloch

There will be 3 contact hours per week comprising one lecture, one tutorial and one hour which may be used as an additional lecture some weeks, and as an interactive class on some weeks. The content will be delivered in part by lectures, and in part by asynchronous videos. There will be weekly exercise sheets to study independently and review within tutorial classes.
 

Coursework: Single piece of take-home coursework, weighting 20%

End of semester examination: weighting  80%

Feedback tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer’s office hour.

The course notes are self-contained, no additional reading is required. The course is based on the following books:

Stewart and Tall, Algebraic Number Theory and Fermat's Last Theorem (recommended)
Jarvis, Algebraic Number Theory (recommended)
Marcus, Algebraic Number Fields (recommended)
Neukirch, Algebraic Number Theory (further)