The minimal periodicity for integral bases of pure number fields
Khai-Hoan Nguyen-Dang
TL;DR
The paper studies the ring of integers in pure number fields $K_a=\mathbb Q(\sqrt[n]{a})$ with $a$ nth-power-free, establishing a sharp local-to-global principle for the integral basis shape in the fixed $\{1,\theta,\dots,\theta^{n-1}\}$ coordinates. It introduces the shape formalism, showing that each local $p$-shape is determined by a single extra $p$-adic digit beyond $p^e$ and that the global shape is periodic with minimal modulus $M(n)=\prod_{p^e\parallel n} p^{e+1}=n\cdot\mathrm{rad}(n)$, thus unifying and sharpening prior results of Gaál–Remete and Jakhar–Khanduja–Sangwan. The results yield practical tools: finite lookup tables for shapes, sharp local index tests at primes dividing $n$, and density/counting statements for shape classes, with explicit quartic and sextic examples. The work provides a coherent local–to–global framework for understanding when and how integral bases in pure fields exhibit periodicity, and it quantifies the minimal information needed from $a$ to determine the shape of the integral basis.
Abstract
Fix $n\ge3$. For the pure field $K_a=\mathbb Q(θ)$ with $θ^n=a$, where $a\neq \pm 1$ is $n$th-power-free, we encode an integral basis in the fixed coordinate $\{1,θ,\dots,θ^{n-1}\}$ 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.