Invariant fields of finite irreducible reflection groups

We prove the following result: If G is a finite irreducible reflection grou p defined over a base field k, then the invariant field of G is purely tran scendental over k, even if \G\ is divisible by the characteristic of k. It is well known that in the above situation the invariant ring is in general not a polynomial ring. So the question whether at least the invariant field is purely transcendental (Noether's problem) is quite natural.