  We consider the notion of finite type-ness of a site introduced by Morel and
Voevodsky, for the \'etale site of a field. For a given field $k$, we
conjecture that the \'etale site of $Sm/k$ is of finite type if and only if the
field $k$ admits a finite extension of finite cohomological dimension. We prove
this conjecture in some cases, e.g. in the case when $k$ is countable, or in
the case when the $p$-cohomological dimension $cd_p(k)$ is infinite for
infinitely many primes $p$.
