Computation of an Integral Basis of Quartic Number Fields

TL;DR

This paper presents a method using Newton polygons to compute p-integral bases for quartic number fields defined by specific polynomials, applicable to all primes p in a systematic way.

Contribution

It introduces a general technique for calculating p-integral bases of quartic fields using Newton polygons, applicable to any prime p and polynomial form.

Findings

Provides a complete method for p-integral basis computation

Applicable to all primes p for quartic fields

Based on Newton polygon techniques

Abstract

In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a quartic number field K defined by an irreducible polynomial $P(X) = X4 + aX + b \in\z[X]$ in methodical and complete generality.