On prolongations of valuations via newton polygons and liftings of polynomials

Abstract

Let v be a real valuation of a field K with valuation ring Rv. Let K(θ) be a finite separable extension of K with θ integral over Rv and F(x) be the minimal polynomial of θ over K. Using Newton polygons and residually transcendental prolongations of v to a simple transcendental extension K(x) of K together with liftings with respect to such prolongations, we describe a method to determine all prolongations of v to K(θ) along with their residual degrees and ramification indices over v. We give an analogue of Ore's Theorem when the base field is an arbitrary rank-1 valued field which extends the main result of [Mathematika, 47 (2000), 173--196].