  We prove a relative decidability result for perfectoid fields. This applies
to show that the fields $\mathbb{Q}_p(p^{1/p^{\infty}})$ and
$\mathbb{Q}_p(\zeta_{p^{\infty}})$ are (existentially) decidable relative to
the perfect hull of $ \mathbb{F}_p(\!(t)\!)$ and $\mathbb{Q}_p^{ab}$ is
(existentially) decidable relative to the perfect hull of $\overline{
\mathbb{F}}_p(\!(t)\!)$. We also prove some unconditional decidability results
in mixed characteristic via reduction to characteristic $p$.
