A generalized fundamental principle

Let upsilon be a rank 1 henselian valuation of a field K having unique exte nsion <(upsilon)over bar> to an algebraic closure (K) over bar of K. For an y subextension L/K of (K) over bar/K, let G(L), Res(L) denote respectively the value group and the residue field of the valuation obtained by restrict ing <(upsilon)over bar> to L. If a is an element of (K) over bar\K define delta(K)(a) = sup {<(upsilon)over bar>(a - c)\c is an element of (K) over b ar, [K(c) : K] < [K(a) : K]}, omega(K) (a) = max {<(upsilon)over bar>(a - a')\a' not equal a runs over K- conjugates of a}. In this paper, it is shown that the constant delta(K)(a) satisfies a princi ple which is similar to the one satisfied by omega(K)(a), namely Krasner's Lemma. The authors prove that, if a, b is an element of (K) over bar are su ch that <(upsilon)over bar>(a - b) > delta(K)(a), then: (i) G(K(a)) subset of or equal to G(K(b)); (ii) Res(K(a)) subset of or equal to Res (K(b)); (iii) [K(a) : K] divides [K(b) : K]. For a is an element of K-sep, they also investigate when the inequality del ta(K)(a) less than or equal to omega(K)(a) becomes equality.