Decidability via the tilting correspondence

We prove a relative decidability result for perfectoid fields. This applies to show that the fields ${\mathbb{Q}}_p(p^{1∕p^{\infty }})$ 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 $p$.