MTH3150 · 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.

Areas of study: Applied mathematics Mathematical statistics Mathematics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

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.

Workload

• Three 1-hour seminars; • One 2-hour applied class (in weeks 2-12) and • 7 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.