Higher Newton polygons and integral bases

Abstract

Let $A$ be a Dedekind domain, $K$ the fraction field, $\p$ a non-zero prime ideal of $A$, and $K_\pp$ the completion of $K$ with respect to the $\p$-adic topology. At the input of a monic irreducible separable polynomial, $f(x)\in A[x]$, Montes algorithm determines the factorization of $f(x)$ over $K_\pp[x]$, and it provides essential arithmetic information about the finite extensions of $K_\pp$ determined by the different irreducible factors. In particular, it can be used to compute $\p$-integral bases of the extension of $K$ determined by $f(x)$ \cite{newapp}. In this paper we present new (and faster) methods 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 Montes algorithm.