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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

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