A generalization of the Grunwald–Wang theorem for <i>n</i>th powers

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 nth power in [Formula: see text] for almost every prime p if and only if A contains a perfect nth power, barring some exceptions when n is even. This generalizes the Grunwald–Wang theorem for nth 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.