# Why is $\mathbb{Q}_p(p^{1/p^\infty})$ a complete topological field?

In Matthias Wulkau's exposition of Scholze's thesis, the term perfectoid field is defined as follows:

Let $K$ be a field endowed with a non-archimedian absolute value $\lvert\cdot\rvert$, and let $\mathcal{O}_K$ and $\mathfrak{m}$ be the closed and open unit balls in $K$, respectively. We say that $K$ is a perfectoid field if $\lvert K^\times\rvert\subset\mathbb{R}_{\ge0}$ is non-discrete, if $\operatorname{char}(\mathcal{O}_K/\mathfrak{m})=p>0$, and if the Frobenius map $$\Phi:\mathcal{O}_K/(p)\to\mathcal{O}_K/(p),\ \ x\mapsto x^p$$ is surjective.

Now, I'm slightly confused by this definition, since I know that $L:=\mathbb{Q}_p(p^{1/p^\infty})$ is meant to be an example of a perfectoid field, however, it would seem to me (by analogy with $\overline{\mathbb{Q}_p}$) that $L$ is not complete, and I'm having trouble seeing why $L$ should be complete.

• I think that $L$ is not perfectoid, precisely because it is not complete. Its completion, however, is perfectoid, – Jesse Silliman Apr 27 '17 at 3:05
• Yeah, I agree, any time you've heard that used as an example, they either said it was the completion of that field, or they forget to mention it. – Will Sawin Apr 27 '17 at 3:11
• Note that since $K = \mathbf{Q}_p(p^{1/p^{\infty}})$ has henselian valuation ring (direct limit of complete discrete valuation rings), its Galois theory is canonically identified with that of its completion in the sense that $E \rightsquigarrow \widehat{K} \otimes_K E$ is an equivalence between the categories of finite etale $K$-algebras and finite etale $\widehat{K}$-algebras; see [2.3.1, 2.4.1-2.4.3] in Berkovich's paper on etale cohomology for non-archimedean spaces in Publ. Math. IHES 78. Making "algebraic approximations" to perfectoid constructions is important in some proofs. – nfdc23 Apr 27 '17 at 7:19
• Note that you do not ask that the field be complete in you boxed statement... – ACL May 5 '17 at 20:11

## 1 Answer

This was alredy answered in the comments, it is the $p$-adic completion of $\mathbb{Q}_p(p^{1/p^\infty})$ that is a perfectoid field.

But here is a reference for completeness