Higher Newton polygons and integral bases

Abstract

Abstract. Let A be a Dedekind domain whose field of fractions K is a global field. Let p be a non-zero prime ideal of A, and Kp the completion of K at p. The Montes algorithm factorizes a monic irreducible separable polynomial f (x) ∈ A[x] over Kp, and it provides essential arithmetic information about the finite extensions of Kp determined by the different irreducible factors. In particular, it can be used to compute a p-integral basis of the extension of K determined by f (x). In this paper we present a new and faster method to compute p-integral bases, based on the use of the quotients of certain divisions with remainder of f (x) that occur along the flow of the Montes algorithm.

Reduced normal form of local integral bases

“…, ν n are known. Such a basis is provided, for instance, by the method of the quotients [6], or the multipliers method [1], both based on the Montes algorithm [5,4].…”

“…We start with the n-reduced poynomial family provided by either method, quotients [6] or multipliers [1]. Let r be the number of pairwise different maximal w-values.…”

“…. , w t the valuations on L extending v. The notion of reduced families of elements in L with respect to the function w : L * −→ Q, w(α) = Min{w i (α) | 1 ≤ i ≤ t} was already introduced in [6] as a tool to prove that certain families of integral elements constitute an A-basis of B.…”

Discriminants of Chebyshev radical extensions

“…Our proof relies on the work of Guàrdia, Montes, and Nart [9,10,11]. In particular, the computation of the index comes from an analysis of specialized Newton polygons, which we describe in Section 4.…”

“…We compute ind(Φ) using a relatively recent algorithm derived by Guàrdia, Montez, and Nart [9,10,11]. Their method employs a more refined variation of the Newton polygon, called the φ-Newton polygon, which captures arithmetic data attached to each irreducible factor φ of Φ.…”