Tame fields and tame extensions

Abstract

Let V be a henselian valuation of any rank of a field K and let V̅ be the extension of V to a fixed algebraic closure K̅ of K. In this paper, it is proved that (K, V) is a tame field, i.e., every finite extension of (K, V) is tamely ramified, if and only if, to each α ∈ K̅\K, there correspondsa ∈ K for which V̅(α − a) ≥ Δ_K(α), where Δ_K(α) = min{V̅(α′ − α)|α′ runs over all K-conjugates of α}. A special case of the previous result, when K is a perfect field of nonzero characteristic was proved in 1995, with the purpose of completing a result of James Ax [S. K. Khanduja,J. Algebra172(1995), 147–151].