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

Contacts

Chief Examiner(s)

Associate Professor Heiko Dietrich

Unit Coordinator(s)

Associate Professor Heiko Dietrich

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 - In-semester assessment
2 - Examination (3 hours and 10 minutes)

Scheduled and non-scheduled teaching activities

Applied sessions
Lectures

Workload requirements

Workload

Learning resources

Required resources
Recommended resources

Other unit costs

Costs are indicative and subject to change.
Miscellaneous items required (Unit Course Reader, Printing, Stationery) - $120.

Availability in areas of study

Applied mathematics
Mathematical statistics
Mathematics
Pure mathematics