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Grunwald–Wang theorem
In algebraic number theory, the Grunwald–Wang theorem is a local-global principle stating that—except in some precisely defined cases—an element ''x'' in a number field ''K'' is an ''n''th power in ''K'' if it is an ''n''th power in the completion of ''K''. For example, a rational number is a square of a rational number if it is a square of a ''p''-adic number for almost all primes ''p''. The Grunwald–Wang theorem is an example of a local-global principle.
It was introduced by , but there was a mistake in this original version that was found and corrected by . The theorem considered by Grunwald and Wang was more general than the one stated above as they discussed the existence of cyclic extensions with certain local properties, and the statement about ''n''th powers is a consequence of this.
== History ==
, a student of Hasse, gave an incorrect proof of the erroneous statement that an element in a number field is an ''n''th power if it is an ''n''th power locally almost everywhere. gave another incorrect proof of this incorrect statement. However discovered the following counter-example: 16 is a ''p''-adic 8th power for all odd primes ''p'', but is not a rational or 2-adic 8th power. In his doctoral thesis written under Artin, Wang gave and proved the correct formulation of Grunwald's assertion, by describing the rare cases when it fails. This result is what is now known as the Grunwald–Wang theorem. The history of Wang's counterexample is discussed in
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