Let $K$ be a field, $K^{sep}$ its separable closure, and $G = Gal(K^{sep}/K)$ the absolute Galois group of $K$. What are some algebraic conditions on $K$ that ensure that $G$ is first-countable? This is equivalent to there being a countable nested sequence of finite-index open subgroups that intersect trivially, which in turn is equivalent to $G$ being metrizable.
This occurs if $K$ is countable, but for the moment I cannot think of other general criteria.