Grunwald-Wang Theorem

The Grunwald-Wang theorem is a local-global principle in algebraic number theory. It states that, except in certain precisely defined exceptional cases, an element of a number field is an nth power in that field if it is an nth power in the completion of the field at all but finitely many primes; for example, a rational number is the square of a rational number if it is the square of a p-adic number for almost every prime p. The theorem was introduced by Wilhelm Grunwald in 1933, but his original version contained an error that was found and corrected by Shianghao Wang in 1948. Grunwald and Wang's original result was actually broader, addressing the existence of cyclic extensions with certain local properties, of which the statement about nth powers is a consequence.