A valued field is said to be algebraic maximal if all its algebraic extension have either a bigger value group or a bigger residue field.
Do you know for which field this is a first order property ? Tanks.
A valued field is said to be algebraic maximal if all its algebraic extension have either a bigger value group or a bigger residue field. Do you know for which field this is a first order property ? Tanks. 


The property of being algebraically maximal is a firstorder property in the language of valued fields, $\{+,,\times, 0, 1, \mid\}$, where `$x\mid y$' if and only if '$v(x)\leq v(x)$'. This is proved in Quelques propriétés des corps values en théorie des modèles (1982), by F.Delon. More specifically, it is proved that


