A Characterization of Finite Tame Extensions

Let v be a henselian valuation of a field K. In this paper itis proved that any finite extension (K', v') of (K, v) is tameif and only if there exists 0 in K' such that v'() = v(Tr_K'/K())using elementary results of valuation theory. A special caseof this result, when the characteristic of the residue fieldof v is p > 0 and (K', v')/(K, v) is an extension of degreep, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math.404 (1990) 1–38). 1991 Mathematics Subject Classification12J10, 12J25, 13A18.