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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.03301 |
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Table of 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 $n$.