Computation of Integral Bases

TL;DR

This paper introduces a faster method for computing p-integral bases in algebraic number theory, utilizing simple multipliers derived from the Montes Algorithm to improve efficiency and often produce triangular bases.

Contribution

It presents a novel, more efficient technique for calculating p-integral bases using simple multipliers from the Montes Algorithm, enhancing speed and basis structure.

Findings

Method significantly faster than previous approaches

Often produces a triangular basis

Applicable to various prime ideals in Dedekind domains

Abstract

Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ we present in this paper a new method to compute a $\mathfrak{p}$-integral basis of the extension of $K$ determined by $f$. Our method is based on the use of simple multipliers that can be constructed with the data that occurs along the flow of the Montes Algorithm. Our construction of a $\mathfrak{p}$-integral basis is significantly faster than the similar approach from $[7]$ and provides in many cases a priori a triangular basis.