6 points, SCA Band 2, 0.125 EFTSL
Undergraduate - Unit
Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.
School of Mathematical Sciences
Rings, fields, ideals, number fields and algebraic extension fields. Coding theory and cryptographic applications of finite fields. Gaussian integers, Hamilton's quaternions. Euclidean Algorithm in rings.
On completion of this unit students will be able to:
- Formulate abstract concepts in algebra;
- Use a variety of proof-techniques to prove mathematical results;
- 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;
- Understand 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;
- Apply the classification of finite fields;
- Generalise known concepts over the integers to other domains, for example, use the Euclidean algorithm or factorisation algorithms in the algebra of polynomials;
- Construct larger fields from smaller fields (field extensions and splitting fields);
- Apply field theory to coding and cryptography; understand the classification of cyclic codes.
NOTE: From 1 July 2019, the duration of all exams is changing to combine reading and writing time. The new exam duration for this unit is 3 hours and 10 minutes.
End of semester examination (3 hours): 60% (Hurdle)
Continuous assessment: 40% (Hurdle)
Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for both the end-of-semester examination and continuous assessment components.
Three 1-hour lectures and one 2-hour applied class per week
See also Unit timetable information