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Invariants of Certain Groups I1)

Published online by Cambridge University Press:  22 January 2016

Takehiko Miyata
Takehiko Miyata*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University
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Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 41 , February 1971 , pp. 69 - 73
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

Footnotes

2)

The author wants to thank Professors H. Hironaka and H. Matsumura who encouraged him constantly during this research. He also wants to thank Professor R. Friedberg and Dr. L. Olson who helped the author in writing English.

1)

This is a part of the author’s thesis written under the guidance of Professor Heisuke Hironaka at Columbia University.

References

[1] Borei, A.; Groupes algébriques linéaires. Annals of Math. (2) 64, 20–82 (1956).Google Scholar
[2] Gaschutz, W., Fixkörper von p-automorphismengruppen rein transzendenter Korpererweiterungen von p-Charakteristik. Math. Z., Vol. 71 (1969).Google Scholar
[3] Kuniyoshi, H.; Certain subfields of rational function fields. Proc. International Sym. on Algebraic Number Theory, 241–243 (1955).Google Scholar
[4] Noether, E.; Gleichungen mit vorgebshriebene Gruppe. Math. Ann. 78, 221–229 (1916).Google Scholar
[5] Serre, J. -P.; Corps locaux. Paris.Google Scholar
[6] Serre, J. -P.; Lie algebras and Lie groupes. Benjamin,Google Scholar