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 fieldif $\lvert K^\times\rvert\subset\mathbb{R}_{\ge0}$ 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.