Tame Fields and Tame Extensions,

LetVbe a henselian valuation of any rank of a fieldKand let be the extension ofVto a fixed algebraic closure ofK. 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 \K, there correspondsa Kfor which (α − a) ≥ ΔK(α), where ΔK(α) = min{ (α′ − α)α′ runs over allK-conjugates of α}. A special case of the previous result, whenKis 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].