On a Problem of Chevalley

Abstract

Recently Prof. Chevalley in Nagoya suggested to the author the following problem: Let k be a field, K 5 = k ( x 1 , x 2 , x 3 , x 4 , x 5 ) be a purely transcendental extension field (of transcendental degree 5) of k , s 5 be the cyclic permutation of x : S 5 X 1 = x 2 s 5 x 2 = x 3 s 5 x 3 = x 4 s 5 x 4 = x 5 s 5 x 5 = x 1 , and let L 5 be the field of invariants of s 5 in K 5 . Is L 5 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].