Henselian valued fields for characteristic $0$: a characterization

Question

Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property:

$K$ is Henselian iff $K=K_v\cap \overline{K}$ where $K_v$ is the completion wrt distance $\vert \ \vert_v$ and $\overline{K}$ is the algebraic closure of $K$.

I'm familar with these equivalent characterization of Henselness. Another property that I know is that if $K$ Henselian and $L/K$ an finite algebraic extension then $v$ extends uniquely to $O_L$. But I don't know how it could help me here.

In other words why every element $a \in K_v \backslash K$ cannot be algebraic over $K$ if $char(K)=0$? Assume $a$ is algebraic. Then $K(a) /K$ is a separable extension of $K$ contained in $K_v$. What is the contradiction? Does something going wrong with uniqueness of extension of $v$ to $K(a)$?

I pretty sure that this possibly not a research question but unfortunately having asked exactly the same question in MSE (meanwhile deleted) I haven't obtain an answer.

Answer 1

In general, i.e. for any valued field $(K,v)$, the implication $K = K_v \cap \overline{K}$ (or more precisely that $(K,v)$ have no immediate algebraic extension, i.e. that $(K,v)$ be algebraically maximal, otherwise you seem to be already assuming that $v$ extends uniquely to $\overline{K}$) implies that $K$ is henselian.

To see this, one can apply Newton's algorithm to find, starting with a Hensel configuration, a pseudo-Cauchy sequence of algebraic type which pseudo-converges to a solution for this configuration. See here, Chapter 3.3. The notions of pseudo-convergence was developped by Ostrowski and then Kaplansky who also introduced the notion of algebraic type for pseudo-Cauchy sequences.

If the residue field has characteristic zero, then the converse implication holds. You can also find a proof in the previous link. I suggest you look at both chapters 3.2 and 3.3 to get a full picture.