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
- know and recognise several constructions of groups
- know and work with group homomorphisms and understand their role in the more general context of mathematics
- work with rings and fields
- investigate and understand when two algebraic structures are isomorphic
- understand and describe several applications of Algebra like encryption based on RSA, the word problem, symmetries and isomorphisms, TDA, HTT
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 non-commutative.
The backbones of the course are constructions for groups and the understanding of structure-preserving maps. As to applications of algebraic structures, the course addresses encryption based on RSA, the word problem, symmetries and isomorphisms in adjacent mathematical domains, topological data analysis, and homotopy type theory. The learning and understanding of abstract algebraic structures will be supported by mandatory Grasple exercises.
Topics covered in this course are:
- Constructions of groups: Integers and integers modulo n, matrix groups, symmetry groups and groups of functions
- Group Theory: group operations, cosets & Lagrange Theorem, group homomorphism, crucial structures in groups
- Rings and Fields
- Applications of Algebra including encryption, data analysis and the recognition of structure-preserving transformations