Grunwald–Wang Theorem

(Theory)

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Theory since 1933
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What is 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 nth power in K if it is an nth power in the completion for all but finitely many primes 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 Wilhelm Grunwald , but there was a mistake in this original version that was found and corrected by Shianghao Wang . 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 nth powers is a consequence of this.

Technology Types

class field theorypropositionstatementtheoremtheorems in algebraic number theorytheory

Synonyms

Grunwald theoremGrunwald-Wang theoremGrunwald-Wang's theoremGrunwald's theorem

Translations

Théorème de Grunwald-Wang (fr)

Tech Info


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