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One-Valued Mappings of Groups into Fields

Published online by Cambridge University Press:  22 January 2016

Katsuhiko Masuda
Katsuhiko Masuda*
Affiliation:
Department of Mathematics, Yamagata University
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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.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 41 - 52
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

1) Tannaka, T., Über den Dualitätssatz der nicht kommutativen topologische Gruppe, Tohoku Math. J. 45(1938)Google Scholar.

Krein, M., On the almost periodic functions on a topological group, C. R. (Doklady) 30(1941)Google Scholar.

2) Cf. Tannaka’s above article.

3) van Kampen, E. R., Almost periodic functions and compact groups, Annals of Math. 37(1936)Google Scholar.

4) Nakayama, T., Construction and characterization of Galois algebras with given Galois group, Nagoya Math. J. 1(1950)Google Scholar.

5) Osima, Cf. M., Note on the Kronecker product of representations of group, Proc. Acad. Japan 17(1941)Google Scholar.

6) An associative, commutative, and semisimple Galois algebra is necessarily absolutely semisimple; cf. H. I.

7) Hasse, H., J. reine angew. Math. 187(1950)Google Scholar.

8) Cf. 6).

9) Nakayama, T., J. reine angew. Math. 189(1951)Google Scholar.

10) Masuda, K., Tôhoku Math. J. 4(1952)Google Scholar.

11) This remark is due to S. Takahasi.

12) We do not use the compactness of either

13) Γ resp. Γ*becomes then by Kampeh’s theorem a complete representative system of all conjugate classes of irreducible unitary continuous representations of G resp. . But in the present article we do not use that fact.

14) We do not explicitly deal with the totality of continuous almost periodic functions of either or G, though resp. coincides really that of resp. G.

15) Here we need apply only classical Weierstrass approximation theorem to concretely defined two real functions of a real variable. But we need apply Neumann’s generalized one and Urysohn’s existence theorem of continuous functions to prove the existence of sufficiently many unitary continuous representations of G, which we supposed in the present article already in § 1, No. 3.