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In this course we have learnt that modern algebra is a study of sets with operations defined on them. As the main example we have started a systematic study of groups. Group theory is one of the most important areas of contemporary mathematics, with applications ranging from physics and chemistry to coding and cryptography. It is also one of the research interests in this school. Further study of groups can be undertaken in the appropriate honours modules.

As our second example, we have given a brief introduction to rings and fields. We have seen that there are some important properties which are very similar to groups. Further courses on rings are also available at the honours level.

Today, groups, rings and fields, along with vector spaces, are regarded as classical algebraic disciplines. There is also a wide variety of newer structures: semigroups, lattices, boolean algebras, etc.

Edmund F Robertson

11 September 2006