Decidability via the tilting correspondence

We prove a relative decidability result for perfectoid fields. This applies to show that the fields ${\mathbb{Q}}_p(p^{1\mathrm{/}{p}^{\mathrm{\infty }}})$ and ${\mathbb{Q}}_p({\zeta }_{p^{\mathrm{\infty }}})$ are (existentially) decidable relative to the perfect hull of ${\mathbb{F}}_p(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$ 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 $p$.