  Let $K$ be a complete discretely valued field with mixed characteristic $(0,
p)$ and imperfect residue field $k_K$. Let $\Delta$ be a finite set. We
construct an equivalence of categories between finite dimensional
$\Bbb{F}_p$-representations of the product of $\Delta$ copies of the absolute
Galois group of $K$ and multivariable \' etale $(\varphi, \Gamma)$-modules over
a multivariable Laurent series ring over $k_K$.
