Timeline for Is any nonarchimedean field containing all roots of unity perfectoid?

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May 13, 2018 at 16:22	comment	added	user124375		The $p$-adic completion of $K(T)$ will include power series such that the coefficients converge to 0 $p$-adically. For example, something like $\sum_0^\infty p^nT^n$ will be there. The valuation ring will depend on the topology you put on $K(T)$. If you put the $p$-adic topology on, it'll be the $p$-adic completion of $\mathcal O_K(T)$. In particular, things like $1/T$ will be in the valuation ring. If you put the $(p,T)$-adic topology on, it'll be the $p$-adic completion of $\mathcal O_K[T]$.
May 12, 2018 at 15:13	comment	added	clueless		Thanks! Can you clarify what you mean by the $p$-adic completion of $K(T)$? In particular, what is the valuation ring?
May 12, 2018 at 5:58	history	edited	Will Sawin	CC BY-SA 4.0	added 27 characters in body
May 12, 2018 at 4:42	review	Low quality posts
May 12, 2018 at 5:58
May 12, 2018 at 4:27	review	First posts
May 12, 2018 at 4:34
May 12, 2018 at 4:24	history	answered	user124375	CC BY-SA 4.0