On extensions generated by roots of lifting polynomials

Let v be a Henselian valuation of any rank of a field K and D its unique prolongation to a fixed algebraic closure (K) over bar of K having value group (G) over bar. For any subfield L of K, let R(L) denote the residue field of the valuation obtained by restricting (v) over bar to L. Using the canonical homomorphism from the valuation ring of v onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, v)-minimal pair (alpha, delta) belonging to (K) over bar x (G) over bar. As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(alpha))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (alpha, delta). If F, F-1 are two such lifted polynomials with coefficients in K having roots theta, theta(1), respectively, then it is proved in the present paper that (v) over bar (K(theta)) = (v) over bar (K(theta(1))), R(K(theta)) = R(K(theta(1))); in case (K, v) is a tame field, it is shown that K(theta) and K(theta(1)) are indeed K-isomorphic.