The minimal periodicity for integral bases of pure number fields

Abstract

Fix $n\ge3$. For the pure field $K_a=\mathbb Q(\theta)$ with $\theta^n=a$, where $a\neq \pm 1$ is $n$th-power-free, 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 $a\bmod p^{\,e+1}$, 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.