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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.

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Could you clarify which formal language you intend? – Joel David Hamkins Sep 18 '12 at 23:28
Thanks for the answer. I didn't think of any language in particular, I'm interested in any example of language in which being algebraic maximal is first order. – nadeau Sep 19 '12 at 7:35

The property of being algebraically maximal is a first-order 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

$(K,V)$ is algebraically maximal, if and only if, for each positive number $n$, every polynomial $P\in K[X]$ of degree not larger than $n$ has the property that, there exists an $x$ such that for all $y$, $v(P(y))\leq v(P(x))$.

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