Computable algebra, general theory and theory of computable fields.

Abstract

algebra is the product of a transition from such concrete systems as the ring of integers or the field of rational numbers to the corresponding, axiomatically defined, abstract systems. In the course of this abstraction process various properties of the concrete system are lost. The field of rational numbers, for example, possesses a natural topology, an ordering, and furthermore the field operations, that is addition and multiplication of rationals, are effectively computable functions. None of these properties enters into the axiomatic definition of a field. Various attempts were made to reincorporate some of these features into the study of algebraic systems. In topological algebra we study groups endowed with a topology subject to the condition that the algebraic operations are continuous functions with respect to this topology. The basic definitions and the concepts studied are a blend of algebra and topology. Thus we are most interested in those homorphisms which are also continuous mappings.