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Overview

Rings, fields, ideals, number fields and algebraic extension fields. Coding theory applications of finite fields. Gaussian integers, Hamilton's quaternions. Euclidean Algorithm in rings.

Offerings

S2-01-CLAYTON-ON-CAMPUS

Rules

Enrolment Rule

Contacts

Chief Examiner(s)

Dr Tomasz Popiel

Unit Coordinator(s)

Dr Tomasz Popiel

Learning outcomes

On successful completion of this unit, you should be able to:
1.

Formulate abstract concepts in algebra;

2.

Use a variety of proof-techniques to prove mathematical results;

3.

Work with the most commonly occurring rings and fields: integers, integers modulo n, matrix rings, rationals, real and complex numbers, more general structures such as number fields and algebraic extension fields, splitting fields, algebraic integers and finite fields;

4.

Demonstrate understanding of different types of rings, such as integral domains, principal ideal domains, unique factorisation domains, Euclidean domains, fields, skew-fields; amongst these are the Gaussian integers and the quaternions - the best-known skew field;

5.

Demonstrate understanding of the classification of finite fields;

6.

Generalise known concepts over the integers to other domains, for example, use the Euclidean algorithm or factorisation algorithms in the algebra of polynomials;

7.

Construct larger fields from smaller fields (field extensions and splitting fields);

8.

Apply field theory to coding theory and understand the classification of cyclic codes.

Teaching approach

Active learning

Assessment

1 - Continuous assessment

2 - Final assessment - Exam (3 hours and 10 minutes)

Scheduled and non-scheduled teaching activities

Applied sessions

Seminars

Workload requirements

Workload

Learning resources

Required resources

Recommended resources

Availability in areas of study

Applied mathematics
Mathematical statistics
Mathematics
Pure mathematics