Integral bases and monogenity of pure fields

TL;DR

This paper investigates the structure of integral bases in pure fields generated by nth roots of square-free integers, revealing periodic patterns and providing explicit bases and index forms to analyze monogenity.

Contribution

It introduces a periodic structure of integral bases in pure fields, explicitly describes these bases for n=3 to 9, and develops a new technique using index forms to study monogenity.

Findings

Integral bases are periodic in m with period n^2 for 3≤n≤9.

Explicit integral bases and index forms are provided for n=3,4,5,6,8.

New methods are developed to determine monogenity of pure fields.

Abstract

Let $m$ be a square-free integer ($m\neq 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\leq n\leq 9$ we show that the period length is $n^2$. We explicitly describe the integral bases, and for $n=3,4,5,6,8$ we explicitly calculate the index forms of $K$. This enables us in many cases to characterize the monogenity of these fields. Using the explicit form of the index forms yields a new technic that enables us to derive new results on monogenity and to get several former results as easy consequences. For $n=4,6,8$ we give an almost complete characterization of the monogenity of pure fields.