One-Valued Mappings of Groups into Fields

The aim of this article is to investigate algebraic nature of systems of one-valued mappings of given group into given field and to apply it to the theory of Galois algebras and duality of compact T0-groups. The results obtained in the following are those; factor systems of Galois algebras with finite Galois groups are defined without any restrictions on the orders of Galois groups and the coefficient fields, a necessary and sufficient condition for them to be associated with Galois fields is obtained, dualities of finite groups are obtained very simply without any restrictions for coefficient field of representations, and Tannaka’s duality of compact T0-groups is proved without the use of the compactness of Tannaka representation groups of representations of compact T- groups and the use of Kampen’s theorem.