Does anyone know of an elementary proof that an algebraically maximal field is Henselian (ie one that does not assume knowledge of henselizations)?
Definitions: We say a valued field $(K,v)$ is algebraically maximal if it has no proper algebraic intemediate extensions, that is any algebraic valued field extension of $(K,v)$ either increases the value group or the residue field.
We say a valued field is Henselian if if satisfies Hensels lemma, for instance any polynomial $X^n+X^{n-1}+a_{n-2}X^{n-2}+...+a_0$ with $v(a_{n-2}),...,v(a_0)>0$ has a root $b$ in $K$ with $v(b)\geq 0$.