Square-free OM computation of global integral bases

TL;DR

This paper extends the OM algorithm from prime-based to integer-based computations, enabling the calculation of global integral bases in number fields without prior discriminant factorization.

Contribution

It introduces a novel adaptation of OM techniques for positive integers, facilitating global integral basis computation without discriminant factorization.

Findings

Algorithm computes global integral bases efficiently.

No need for prior discriminant factorization.

Applicable to a wide class of number fields.

Abstract

For a prime $p$, the OM algorithm finds the $p$-adic factorization of an irreducible polynomial $f\in\mathbb{Z}[x]$ in polynomial time. This may be applied to construct $p$-integral bases in the number field $K$ defined by $f$. In this paper, we adapt the OM techniques to work with a positive integer $N$ instead of $p$. As an application, we obtain an algorithm to compute global integral bases in $K$, which does not require a previous factorization of the discriminant of $f$.