Master of Mathematics and Physics (MMath&Phys)

MMath&Phys Mathematics and Physics

A diverse, varied course where you can draw on the combined expertise of three centres of University excellence.
  • Duration: 4 years
  • Year of entry: 2025
  • UCAS course code: FG3C / Institution code: M20
  • Key features:
  • Study abroad
  • Scholarships available
  • Accredited course

Full entry requirementsHow to apply

Fees and funding

Fees

Tuition fees for home students commencing their studies in September 2025 will be £9,250 per annum. Tuition fees for international students will be £36,500 per annum. For general information please see the undergraduate finance pages.

Policy on additional costs

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).

Scholarships/sponsorships

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages .

Course unit details:
Rings & Fields

Course unit fact file
Unit code MATH21112
Credit rating 10
Unit level Level 2
Teaching period(s) Semester 2
Available as a free choice unit? Yes

Overview

The unit covers
- properties of the integers including the division theorem, greatest common divisor and prime factorisation
- definition of a ring and subrings with standard examples including number rings, modular rings, matrix rings and polynomial rings
- special types of rings and elements, including domains, division rings, fields, zero divisors, units and nilpotent elements
- homomorphisms and isomorphisms of rings
- ideals of rings
- quotient rings and isomorphism theorems
- polynomial rings and factorisation
- constructing roots of polynomials, Kronecker’s Theorem and extension fields.

Pre/co-requisites

Unit title Unit code Requirement type Description
Groups and Geometry MATH21120 Co-Requisite Compulsory
MATH21112 CO-REQUISITE

Aims

This unit aims to provide an introduction to the algebraic structures of rings and fields, describing the quotient structure and its connection with homomorphisms of rings and presenting important examples with particular emphasis on polynomial rings.

Learning outcomes

On the successful completion of the course, students will be able to:

  • Describe and apply properties of primes and division in the integers.
  • Define rings, domains and fields and describe standard examples.
  • Recognise and construct special types of elements in rings and fields.
  • State and recall proofs of properties of rings and ring homomorphisms and apply these to standard examples.
  • Recognise ideals and use ideals to construct factor rings.
  • Describe properties of polynomial rings and calculate factors and greatest common divisors of polynomials over a field.
  • Describe Kronecker’s Theorem and use it to construct roots of polynomials and field extensions.
     

Assessment methods

Method Weight
Other 20%
Written exam 80%

Coursework

40 minutes test mid-semester    Written feedback on scripts and model solutions within 2-3 weeks     20%


Exam

2 hours    General feedback after exam results are released.    80%

Feedback methods

Coursework
Written feedback on scripts and model solutions within 2-3 weeks   
Exam   General feedback after exam results are released.  

Study hours

Scheduled activity hours
Lectures 24
Practical classes & workshops 5
Tutorials 6
Independent study hours
Independent study 65

Teaching staff

Staff member Role
Radha Kessar Unit coordinator
Louise Walker Unit coordinator

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