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dbr:List_of_incomplete_proofs
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dbr:Grunwald–Wang_theorem
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yago:Message106598915 yago:Statement106722453 yago:Proposition106750804 yago:Theorem106752293 yago:Abstraction100002137 yago:Communication100033020 yago:WikicatTheoremsInAlgebraicNumberTheory
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Théorème de Grunwald-Wang Grunwald–Wang theorem
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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. En théorie algébrique des nombres, le théorème de Grunwald-Wang est un exemple de principe local-global, selon lequel — hormis dans certains cas précisément identifiés — un élément d'un corps de nombres K est une puissance n-ième dans K si c'est une puissance n-ième dans le complété Kp pour presque tout idéal premier p de OK (c'est-à-dire pour tous sauf un nombre fini). Par exemple, un rationnel est le carré d'un rationnel si c'est le carré d'un nombre p-adique pour presque tout nombre premier p.
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Wilhelm Grunwald Peter Roquette Shianghao Wang
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s+1
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Peter Shianghao Wilhelm
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Roquette Grunwald Wang
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Some days later I was with Artin in his office when Wang appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself... Of course he [Artin] was astonished, as were all of us students, that a famous theorem with two published proofs, one of which we had all heard in the seminar without our noticing anything, could be wrong.
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John Tate, quoted by
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30.0
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1933 1948 2005
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section 5.3
dbo:abstract
En théorie algébrique des nombres, le théorème de Grunwald-Wang est un exemple de principe local-global, selon lequel — hormis dans certains cas précisément identifiés — un élément d'un corps de nombres K est une puissance n-ième dans K si c'est une puissance n-ième dans le complété Kp pour presque tout idéal premier p de OK (c'est-à-dire pour tous sauf un nombre fini). Par exemple, un rationnel est le carré d'un rationnel si c'est le carré d'un nombre p-adique pour presque tout nombre premier p. Il a été introduit par (de) en 1933, mais une erreur dans cette première version fut détectée et corrigée par (en) en 1948. 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.
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