On extensions generated by roots of lifting polynomials

Abstract

Let v be a Henselian valuation of any rank of a field K and ν̅ its unique prolongation to a fixed algebraic closure K̅ of K having value group G̅. For any subfield L of K̅ ,let R(L) denote the residue field of the valuation obtained by restricting ν̅ 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 (α, δ) belonging to K̅ × G̅. As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (α, δ). If F, F ₁ are two such lifted polynomials with coefficients in K having roots θ, θ₁, respectively, then it is proved in the present paper that ν̅(K(θ)= ν̅ (K(θ₁)),R(K(θ))= R(K(θ₁))in case (K, v) is a tame field, it is shown that K(θ) and K(θ1) are indeed K-isomorphic.