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Prof Martin Anthony

This course is available on the BSc in Mathematics and Economics, BSc in Mathematics with Economics and BSc in Mathematics, Statistics and Business. This course is available with permission as an outside option to students on other programmes where regulations permit and to General Course students.

MA103 Introduction to Abstract Mathematics and MA211 Algebra and Number Theory are essential pre-requisites.

The aim of the course is to continue the study of abstract algebraic structures and show how these structures can be used to solve concrete problems. There are three strands: Group actions; Rings, polynomials and fields; Applications, including coding and cryptography. Group actions; revision of permutation groups; orbits and stabilizers, the orbit-stabilizer theorem; applications to counting problems. Rings, polynomials and fields: revision of rings; quotient rings; polynomial rings and the Euclidean algorithm for polynomials; irreducible polynomials and factorisation of polynomials. fields; fields as quotients of polynomial rings; construction and properties of finite fields: Applications: Designs and orthogonal latin squares ; Error-correcting codes, including linear codes, cyclic codes and perfect codes; cryptography.

This course is delivered through a combination of classes and lectures totalling a minimum of 30 hours across Lent Term. This year, apart from pre-recorded lecture videos, there will be a weekly live online session of an hour. Depending on circumstances, classes might be online.

Written answers to set problems will be expected on a weekly basis.

Lecture notes will be provided. No additional reading is required, but the following books are recommended for further reading.

Introduction to Algebra, Peter J Cameron (OUP 2007);

Codes, N.L. Biggs (Springer, 2008).

Exam (100%, duration: 2 hours) in the spring exam period.