A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers

TL;DR

The paper extends the Grunwald-Wang theorem from a single rational to finite subsets $A$ of $\mathbb{Q}$ with $|A|\le q$, where $q$ is the smallest prime dividing $n$, showing $A$ contains a $n^{th}$ power in $\mathbb{Q}_{p}$ for almost all primes $p$ iff $A$ contains a perfect $n^{th}$ power, except for a finite number of even-$n$ cases. It provides a tight bound by proving the optimality of $q$ and develops an elementary, hyperplane-based framework that connects local $n^{th}$-power conditions with covering properties in $\mathbb{F}_{q}^{s}$, mediated by Skałba and Schinzel–Skałba type results. The approach also recovers the classical Grunwald-Wang theorem when $|A|=1$ and yields explicit obstructions for even $n$, while supplying constructions that demonstrate the bound’s optimality. Overall, the work deepens the local-global understanding of $n^{th}$-power phenomena in subsets of $\mathbb{Q}$ and extends Grunwald-Wang to a broader, combinatorial setting with sharp cardinality thresholds.

Abstract

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 $n$.

Theorems & Definitions (15)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4
• Remark 1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 3.1
• proof