(Joint work with S. Kumar.) Let v be a real valuation of a field K with valuation ring R_v. Let K(theta) be a finite separable extension of K with theta integral over R_v, and F(x) be the minimal polynomial of theta 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(theta) 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, vol.47 (2000), 173-196.