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.



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


* Matthias Wulkau, [Review of perfectoid fields](https://www.mathi.uni-heidelberg.de/~G.QpAsPi1geom/manuscripts/PF.pdf) (2015) [Remark 1]


