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