The minimal periodicity for integral bases of pure number fields

Abstract

Fix $n\ge 3$. For the pure field $K_a=\mathbb{Q}(\theta )$ with ${\theta }^n=a$, where $a\ne \pm 1$ is $n$th-power-free, $x^n-a$ is irreducible over $\mathbb{Q}$, and Hypothesis~\textup{(\textbf{H})} holds, we encode an integral basis in the fixed coordinate $\{1,\theta ,\dots ,{\theta }^{n-1}\}$ by its \emph{shape}. We prove a sharp local-to-global principle: for each $p^e\parallel n$, the local shape at $p$ is determined by $a\mathrm{mod}{p}^{e+1}$, and this precision is optimal. Moreover, the global shape is periodic with minimal modulus \[ M(n)=\prod_{p^e\parallel n}p^{\,e+1}=n\cdot\mathrm{rad}(n), \] providing many applications in the understanding integral bases of pure number fields.

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