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I'm reading Tate's paper about $p$-divisible groups. In Chapter $(2.4)$ he asserts that if you take $R$ a complete DVR with residue field $k$ of characteristic $p>0$, $K$ its field of fractions, $L$ the completion of an algebraic extension of $K$ and $S$ the ring of integers in $L$, then $S$ is a complete rank $1$ valuation ring, but with a not necessarily discrete valuation. I'm essentially looking for a counterexample of this fact. I know (see Froelich "Local fields" Prop.2) that, if the extension is finite, then the ring of integers inside $L$ is a DVR, so I'm looking at infinite extensions of local fields. Particularly, I hope that the ring of integers inside $\mathbb{C}_{p}$, the completion of the algebraic closure of $\mathbb{Q}_{p}$ is not a DVR, but I really don't know how to prove this fact. Thanks for any suggestion.

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Pretty sure $L=\mathbb{Q}_p(p^{1/\infty})=\mathbb{Q}_p(\sqrt[n]{p\vphantom{b}}:n\geq 2)$ is a counterexample: letting $v$ be the $p$-adic valuation on $\mathbb{Q}_p$, it would necessarily extend by $v(\sqrt[n]{p\vphantom{b}})=\frac{1}{n}$, making the image of $S^\times$ under $v$ equal to the group of positive rationals $\mathbb{Q}_{>0}$, which is not a discrete group.

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