Integral basis of pure fields with square-free parameter

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Abstract

Let $m\neq0,\pm1$ and $n\geq 2$ be integers. The ring of algebraic integers of the pure fields of type $\mathbb{Q}(\sqrt[n]{m})$ is explicitly known for $n=2,3,4$. It is well known that for $n=2$, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of $m$ modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.\\ In this paper we explicitly give an integral basis of the field $\mathbb{Q}(\sqrt[n]{m})$, where $m\neq\pm1$ is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of $\mathbb{Q}(\sqrt[n]{m})$ is repeating periodically in $m$ with period length depending on $n$.