Reduction in the Rationality Problem for Multiplicative Invariant Fields

For a faithful ZG lattice A and a field K on which the group G acts by field automorphisms, let R be the normal subgroup generated by the elements of G which act trivially on K and act as reflections on A. We prove that the rationality of the multiplicative invariant field K(A)G over K(AR)G is equivalent to the rationality of K(A)ΩG over K (AR)ΩG where ΩG is a particular subgroup of G such that G /R ΩG. We then use this reduction result to prove that K(A)G is rational over K where G is the automorphism group of a crystallographic root system Ψ, G acts trivially on K and A is any lattice on the space QΨ.