I read the following passage in [Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves](https://arxiv.org/abs/1506.00742) 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 completions of 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.



How many untilts?
This specific question is probably not addressed in the literature; let's try to figure it out!


Let $K$ be an algebraic extension of $\mathbb Q\_p$ such that the tilt of $\widehat{K}$ is isomorphic to $\mathbb F\_p((t^{1/p^\infty}))$. We can observe the following:


1. Tilting preserves residue fields, so necessarily $K$ has residue field $\mathbb F\_p$, i.e. is totally ramified over $\mathbb Q\_p$.
2. Tilting preserves value groups, so the value group of $K$ is isomorphic to $\mathbb Z[\tfrac 1p]$. In particular, there can only be a finite amount of ramification of degree prime to $p$.


In particular, $K$ is pro-$p$ and totally ramified of infinite degree over a finite totally tamely ramified extension $K\_0$ of $\mathbb Q\_p$. Conversely, all such $K$ have perfectoid completion $\widehat{K}$.


As SashaP comments below, these conditions are however not yet sufficient for $\widehat{K}^\flat$ to be isomorphic to $\mathbb F\_p((t^{1/p^\infty}))$. For example, one can find such an extension $K/\mathbb Q\_p$ for which $\mathrm{Gal}\_K$ maps isomorphically to the tame quotient of $\mathrm{Gal}\_{\mathbb Q\_p}$; but the tame quotient of $\mathrm{Gal}\_{\mathbb Q\_p}$ cannot be isomorphic to $\mathrm{Gal}\_{\mathbb F\_p((t))}$. It seems to be an interesting question to isolate those $K$ which have tilt isomorphic to $\mathbb F\_p((t^{1/p^\infty}))$!


One definitely gets examples by taking any tower $K\_0=\mathbb Q\_p$, $K\_1$, $K\_2$, ... such that each $K\_{i+1}/K\_i$ is a degree $p$ extension given by extracting a $p$-th root of a uniformizer. At each step, this gives finitely many distinct choices, so there are at least $2^{\aleph\_0}$ such extensions. This is also an evident upper bound, so this gives an answer to b).


