About Henselization

Question

I have some question about Henselization of valued field. If $(K_{1}, \nu_{1})$ is a Henselization of valued field $(K, \nu)$. Which one is true.

1. $K_{1}/K$ is an algebraic extension.
2. $K_{1}/K$ is a transcendental extension.

If $[K(a) : K] = n$ then when $[K_{1}(a): K] = n $ or $< n$. Thanks

Answer 1

A henselization is the a subextension of a separable closure $K^s$ of $K$, so it is always algebraic (extend the valuation $\nu$ to some valuation $\nu_s$ on $K^s$ and take the invariants by the decomposition group at $\nu_s$; you should find this in Engler & Prestel).

You second question doesn't make sense. You probably meant $[K_1(a): K_1]$. Its degree is always at most $n$ (for any field extension $K_1/K$), it can be $1$ if $a\in K_1$ and but can also be $n$.