On a Problem of Chevalley

Abstract

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].