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

TL;DR

This paper generalizes the Grunwald-Wang theorem to finite subsets of rational numbers, establishing conditions under which such sets contain $n^{th}$ powers in local fields for almost all primes, and proves the optimality of the subset size bound.

Contribution

It extends the classical theorem from individual rational numbers to finite sets, providing a sharp bound on subset size for the existence of $n^{th}$ powers locally.

Findings

Finite subsets of rationals of size at most $q$ contain $n^{th}$ powers in $Q_p$ for almost all primes $p$ if and only if the set contains a perfect $n^{th}$ power.

The bound $q$ is proven to be optimal for all $n$.

The result generalizes the classical Grunwald-Wang theorem to a broader setting.

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 $n$.