The minimal periodicity for integral bases of pure number fields

Fix $n\ge 3$. For the pure field $K_a=\mathbb{Q}(\theta )$ with ${\theta }^n=a$, where $a\mathrm{\ne }\pm 1$ 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 $p^e\text{\hspace{0.17em}⁣}\parallel n$, the local shape at $p$ is determined by $a{p}^{e+1}$, and this precision is optimal. Moreover, the global shape is periodic with minimal modulus $$M(n)=\prod _{p^e\parallel n}{p}^{e+1}=n\cdot \mathrm{r}\mathrm{a}\mathrm{d}(n),$$ providing many applications in the understanding integral bases of pure number fields.