TAME FIELDS AND TAME EXTENSIONS

Let V be a henselian valuation of any rank of a held K and let V be th e extension of V to a fixed algebraic closure (K) over bar of K. In th is 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 alpha is an element of (K) over bar \ K,there corresponds a is an element of K for which (V) over bar(alpha -alpha) greater than or equal to Delta (K)(alpha a), where Delta(K)(alpha)= min{(V) over bar(alpha -alpha)\al pha a' runs Over all K-conjugates of alpha). A special case of the pre vious 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. Algebra 172 (1995), 147-151]. (C) 1998 Academic Pres s.