MTH4150 - Algebra 2: Rings and fields

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

Notes

This Level 4 unit and its Level 3 counterpart MTH3150 share the same core content and learning activities such as seminars and applied classes. However, studies at Level 4 are distinguished from those at Level 3 by a deeper understanding of mathematical theories and their applications, higher levels of critical thinking, and greater autonomy in learning.

Learning outcomes

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

- Formulate and critically analyse abstract concepts in algebra.

Apply a wide range of proof techniques to prove complex mathematical results

Master the manipulation and application of the most common rings and fields: integers, integers modulo n, matrix rings, rationals, real and complex numbers, as well as more general structures such as number fields, algebraic extension fields, splitting fields, algebraic integers, and finite fields.

Exhibit an in-depth understanding of different types of rings, such as integral domains, principal ideal domains, unique factorisation domains, Euclidean domains, fields, and skew-fields, amongst these are the Gaussian integers and quaternions, the most well-known skew-fields.

Demonstrate understanding of the classification of finite fields.

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

Construct larger fields from smaller fields (field extensions and splitting fields), demonstrating a deep understanding of these processes.

Apply field theory to coding theory and demonstrate understanding of the classification of cyclic codes

Teaching approach

Active learning

Active learning will occur in seminars and applied classes: regular written assignments will assist you in your revision and learning process; it will teach you to apply the theory discussed in the seminars to explicit problems in algebra. The applied classes will foster group work and discussions: you will collaborate in small groups to work on selected problems. This will foster team-working skills, problem solving skills, and mathematical communication skills (written and oral).

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

Typed skeleton notes

This is a typed version of the lectures notes, but without proofs and significant examples.

Handwritten lecture notes

These are lecture notes that will be prepared live during the lectures; the pdf will be uploaded immediately after each lecture.

Lecture recordings

Echo360 recordings will be provided.

Additional material

The lecturer will upload supplementary material (non-examinable) to cater for those of you who are interested in further studies.

Recommended resources

Availability in areas of study

Applied mathematics
Mathematical statistics
Mathematics
Pure mathematics