  We prove that the theory of a Henselian valued field of characteristic zero,
with finite ramification, and whose value group is a $Z$-group, is
model-complete in the language of rings if the theory of its residue field is
model-complete in the language of rings. We apply this to prove that every
infinite algebraic extension of the field of $p$-adic numbers $\Bbb Q_p$ with
finite ramification is model-complete in the language of rings. For this, we
give a necessary and sufficient condition for model-completeness of the theory
of a perfect pseudo-algebraically closed field with pro-cyclic absolute Galois
group.
