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 $n$.
Categories: math.NT