Multivariable (φ, Γ)-modules and products of Galois groups

We show that the category of continuous representations of the d th direct power of the absolute Galois group of Q(p) on finite dimensional F-p-vector spaces (resp. finitely generated Z(p)-modules, resp. finite dimensional Q(p)-vector spaces) is equivalent to the category of etale (phi, Gamma)- modules over a d-variable Laurent-series ring over F-p (resp. over Z(p), resp. over Q(p)).