A generalization of a theorem of Ore

On prolongations of rank one discrete valuations

“…Ore when ν = ν p in [10] and developed by J. Guardia, J. Montes, and E. Nart in [4]. This notion was generalized to any discrete rank one valuation by D. Cohen, A. Movahhedi, and A. Salinier in their paper [1], and later by B. Jhorar and S. Khanduja in their paper [8], as the polygonal path formed by the lower edges along the convex hull of the points (i, u i ), u i < ∞, in the Euclidean plane, where u i = ν(a i (X)). Geometrically, the ϕ-Newton polygon is represented by the process of joining all segments with an appropriate initial point with increasing slopes λ 0 < λ 1 < • • • < λ g when calculated from left to right.…”

Discriminant and Integral basis of sextic fields defined by x^6+ax+b

“…Using (15), we shall prove that f (x) is 2-regular with respect to φ 1 , φ 2 . Substituting for δ in f (δ) = δ 6 + aδ + b, we have…”

“…Let N denote the number of points with positive integral coordinates lying on or below the φ-Newton polygon of F (x) away from the vertical line passing through the last vertex of this polygon. As in [15], the φ-index of F (with respect to p) is defined to be N deg φ(x) and will be denoted by i φ (F ).…”

“…With the above notation, we shall use the theorem stated below originally proved by Ore (see [15,Theorem 1.2], [21] ). Theorem 3.B.…”

“…With the above notations, we state below a celebrated theorem known as the Theorem of Index of Ore (cf. [15,Theorem 1.4], [21]).…”