Abstract
LetVbe a henselian valuation of any rank of a fieldKand letVbe the extension ofVto a fixed algebraic closureKofK. 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\\K, there correspondsa∈Kfor whichV(α−a)≥ΔK(α), where ΔK(α)=min{V(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].
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