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On a Problem of Chevalley

Published online by Cambridge University Press:  22 January 2016

Katsuhiko Masuda
Katsuhiko Masuda*
Affiliation:
Department of Mathematics, Yamagata University
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Recently Prof. Chevalley in Nagoya suggested to the author the following problem: Let k be a field, K5 = k(x1, x2, x3, x4, x5) be a purely transcendental extension field (of transcendental degree 5) of k, s5 be the cyclic permutation of x: S5X1 = x2s5x2 = x3s5x3 = x4s5x4 = x5s5x5 = x1, and let L5 be the field of invariants of s5 in K5. Is L5 then purely transcendental over k or not? When the characteristic p of k is not equal to 5, it is answered in the following positively. When the characteristic p of k is equal to 5, it is answered also positively by Mr. Kuniyoshi’s result in [2].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 8 , February 1955 , pp. 59 - 63
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

[1] Hasse, H.: Invariante Kennzeichnung Galoisschen Körper mit vorgegebener Galoisgruppe, Crelle J., 187 (1949).Google Scholar
[2] Kuniyoshi, H.: On a problem of Chevalley, the present volume of this Journal.Google Scholar
[3] Masuda, K.: One valued mappings of groups into fields, Nagoya Math. J., 6 (1953).CrossRefGoogle Scholar
[4] Weil, A.: Foundations of Algebraic Geometry, New York (1946).CrossRefGoogle Scholar