
After completion of the course, the student is able to
 Apply the (extended) Euclidean algorithm to integers and polynomials.
 recognise and apply properties of groups
 to derive the symmetry group of a given set and to calculate the number of orbits with the aid of Burnside’s Theorem.
 work with ring and integral domains
 recognise under what conditions an integral domain is a field
 to work with the concept of vector space over an arbitrary field
 work with the classification of finite fields
 work with simple examples of linear code
 investigate and understand when two algebraic structures are isomorphic
 use and explain all components of RSA public key encryption, including all algorithmic underpinnings


This course provides an introduction to abstract algebra and some of its applications. Three families of algebraic structures are introduced and studied: groups, rings and fields. Roughly speaking groups are generalisations of the set of integers equipped with addition. Rings are generalisations of the set of integers equipped with both addition and multiplication. Fields are generalisations of the set of rational numbers equipped with addition, multiplication and division by nonzero numbers. The generalisations go as far as finite structures and structures in which the operation(s) are noncommutative. The backbones of the course are permutations, finite fields. Permutations and symmetries are used in certain counting problems, such as in how many essentially different ways, taking into account the symmetries, can we paint the faces of a cube using three colours. The theory of groups and rings is used to analyse a specific public key crypto system known as RSA. Also the algorithmic principles behind RSA are treated, e.g. the Euclidean algorithm and modular exponentiation. As an application of the theory of finite fields, a brief introduction to finite codes is provided.
Topics covered in this course are:
 Integers and integers module n
 Group Theory, Orbits, Stabilizers
 Cosets & Lagrange Theorem
 Counting problems: Burnside’s theorem
 Rings and Fields
 Polynomial Rings
 Finite Fields
 Linear codes
 RSA Public Key Encryption





Bachelor Applied Mathematics 
  Required materialsBookContemporary Abstract Algebra, Joseph A. Galian, 8th ed., Cengage Learning, 2012. ISBN: 9781133599708 

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