Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v$. 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.

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.

Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v$. I'm looking for a proof of following characterization of Henselian property:

$K$ is Henselian iff $K_v\cap \overline{K}$ 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.

Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v$. 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.