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 $p$.

中文翻译：

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通过倾斜对应的可判定性

我们证明了完美类场的相对可判定性结果。这适用于显示字段${\mathbb{Q}}_p（p^{1∕p^{\mathrm{无穷大}}}）$和${\mathbb{Q}}_p（{\delta }_{p^{\mathrm{无穷大}}}）$相对于完美的船体（存在）是可判定的${\mathbb{𝔽}}_p（（t））$和${\mathbb{Q}}_p^{A乙}$相对于完美的船体是（存在的）可判定的${\stackrel{￣}{\mathbb{𝔽}}}_p（（t））$。我们还通过简化特征证明了混合特征中的一些无条件可判定性结果$p$。