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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

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Do you already know for some reason that only one or the other can occur? If so, then think of the special case where $K$ is already Henselian. –  Matt Jan 21 '13 at 7:14
    
@ Matt, Thanks. I am just reading about the Henselization and I have no idea about those. –  Rajesh Jan 21 '13 at 16:28
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up vote 4 down vote accepted

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 $n$.

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