The minimal periodicity for integral bases of pure number fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912808977825792 |
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| author | Nguyen-Dang, Khai-Hoan |
| author_facet | Nguyen-Dang, Khai-Hoan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The minimal periodicity for integral bases of pure number fields Nguyen-Dang, Khai-Hoan Number Theory 11R04, 11R21 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. |
| title | The minimal periodicity for integral bases of pure number fields |
| topic | Number Theory 11R04, 11R21 |
| url | https://arxiv.org/abs/2509.09457 |