A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915205891489792 |
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| author | Mishra, Bhawesh |
| author_facet | Mishra, Bhawesh |
| contents | Let $n$ be a natural number greater than $2$ and $q$ be the smallest prime dividing $n$. We show that a finite subset $A$ of rationals, of cardinality at most $q$, contains a $n^{th}$ power in $\mathbb{Q}_{p}$ for almost every prime $p$ if and only if $A$ contains a perfect $n^{th}$ power, barring some exceptions when $n$ is even. This generalizes the Grunwald-Wang theorem for $n^{th}$ powers, from one rational number to finite subsets of rational numbers. We also show that the upper bound $q$ in this generalization is optimal for every $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers Mishra, Bhawesh Number Theory 11A15, 11A07, 11R37 Let $n$ be a natural number greater than $2$ and $q$ be the smallest prime dividing $n$. We show that a finite subset $A$ of rationals, of cardinality at most $q$, contains a $n^{th}$ power in $\mathbb{Q}_{p}$ for almost every prime $p$ if and only if $A$ contains a perfect $n^{th}$ power, barring some exceptions when $n$ is even. This generalizes the Grunwald-Wang theorem for $n^{th}$ powers, from one rational number to finite subsets of rational numbers. We also show that the upper bound $q$ in this generalization is optimal for every $n$. |
| title | A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers |
| topic | Number Theory 11A15, 11A07, 11R37 |
| url | https://arxiv.org/abs/2408.03301 |