I read the following passage in Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves by Kedlaya-Temkin: "One can construct many algebraic extensions of $\mathbb{Q}_p$ whose completions $K$ tilt to the completed perfect closure of a power series field over $\mathbb{F}_p$." They then give the two classical examples of the $p$-adic completions of $\mathbb{Q}_p(p^{1/p^{\infty}})$ and $\mathbb{Q}_p(\zeta_{p^{\infty}})$.

This made me wonder:

**Questions**: (*a*) What other algebraic extensions of $\mathbb{Q}_p$ tilt to $\mathbb{F}_p((t^{1/p^{\infty}}))$?
(*b*) How many are there (up to isomorphism)?

To be precise: Question (*b*) asks for the cardinality of the set of untilts of $\mathbb{F}_p((t^{1/p^{\infty}}))$ which are algebraic extensions of $\mathbb{Q}_p$.

I suspect that this question already has an answer in the works of Fargues–Fontaine but my scientific French is too poor to understand if this is the case.