  We investigate the computability-theoretic properties of valued fields, and
in particular algebraically closed valued fields and $p$-adically closed valued
fields. We give an effectiveness condition, related to Hensel's lemma, on a
valued field which is necessary and sufficient to extend the valuation to any
algebraic extension. We show that there is a computable formally $p$-adic field
which does not embed into any computable $p$-adic closure, but we give an
effectiveness condition on the divisibility relation in the value group which
is sufficient to find such an embedding. By checking that algebraically closed
valued fields and $p$-adically closed valued fields of infinite transcendence
degree have the Mal'cev property, we show that they have computable dimension
$\omega$.
