# BSc Mathematics

Year of entry: 2024

## Course unit details:Rings & Fields

Unit code MATH21112 10 Level 2 Semester 2 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
Linear Algebra MATH11022 Pre-Requisite Compulsory
Mathematical Foundation & Analysis MATH11121 Pre-Requisite Compulsory
Groups and Geometry MATH21120 Co-Requisite Compulsory

### 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