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BSc Computer Science and Mathematics / Course details
Year of entry: 2023
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Course unit details:
Rings & Fields
|Unit level||Level 2|
|Teaching period(s)||Semester 2|
|Available as a free choice unit?||Yes|
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.
|Unit title||Unit code||Requirement type||Description|
|Mathematical Foundation & Analysis||MATH11121||Pre-Requisite||Compulsory|
|Groups and Geometry||MATH21120||Co-Requisite||Compulsory|
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.
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.
40 minutes test mid-semester Written feedback on scripts and model solutions within 2-3 weeks 20%
2 hours General feedback after exam results are released. 80%
Written feedback on scripts and model solutions within 2-3 weeks
Exam General feedback after exam results are released.
|Scheduled activity hours|
|Practical classes & workshops||5|
|Independent study hours|
|Radha Kessar||Unit coordinator|
|Louise Walker||Unit coordinator|