Twisted symmetric group actions

Abstract

Let K be any field, let K (x 1 , ... , x n ) be the rational function field of n variables over K, and let S n and A n be the symmetric group and the alternating group of degree n, respectively. For any a ∈ K {0}, define an action of S n on K(x 1 , ... , x n ) by σ · x i = x σ(i) for σ ∈ A n and σ · x i = a/x σ(i) for σ ∈ S n \A n . We prove that for any field K and n = 3, 4, 5, the fixed field K (x 1 , ..., x n ) S n is rational (that is, purely transcendental) over K.