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. The problem is classical but our approach uses new ideas. The paper gives an analogue of Ore’s Theorem when the base field is an arbitrary rank-1 valued field and extends the main result of [S.D. Cohen, A. Movahhedi, A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173–196].