Grunwald–Wang theorem

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 {style d'affichage K_{mathfrak {p}}} for all but finitely many primes {style d'affichage {mathfrak {p}}} of K. Par exemple, 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 (1933), but there was a mistake in this original version that was found and corrected by Shianghao Wang (1948). 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.

Contenu 1 Histoire 2 Wang's counter-example 2.1 An element that is an nth power almost everywhere locally but not everywhere locally 2.2 An element that is an nth power everywhere locally but not globally 3 A consequence of Wang's counter-example 4 Special fields 5 Énoncé du théorème 6 Explanation of Wang's counter-example 7 Voir également 8 Remarques 9 References History 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.

Jean Tate, quoted by Peter Roquette (2005, p.30) Grunwald (1933), a student of Helmut Hasse, gave an incorrect proof of the erroneous statement that an element in a number field is an nth power if it is an nth power locally almost everywhere. George Whaples (1942) gave another incorrect proof of this incorrect statement. However Wang (1948) 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 Wang (1950) written under Emil 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 by Peter Roquette (2005, section 5.3) Wang's counter-example Grunwald's original claim that an element that is an nth power almost everywhere locally is an nth power globally can fail in two distinct ways: the element can be an nth power almost everywhere locally but not everywhere locally, or it can be an nth power everywhere locally but not globally.

An element that is an nth power almost everywhere locally but not everywhere locally The element 16 in the rationals is an 8th power at all places except 2, but is not an 8th power in the 2-adic numbers.

Il est clair que 16 is not a 2-adic 8th power, and hence not a rational 8th power, since the 2-adic valuation of 16 est 4 which is not divisible by 8.

Généralement, 16 is an 8th power in a field K if and only if the polynomial {style d'affichage X^{8}-16} has a root in K. Write {style d'affichage X^{8}-16=(X^{4}-4)(X^{4}+4)=(X^{2}-2)(X^{2}+2)(X^{2}-2X+2)(X^{2}+2X+2).} Ainsi, 16 is an 8th power in K if and only if 2, −2 or −1 is a square in K. Let p be any odd prime. It follows from the multiplicativity of the Legendre symbol that 2, −2 or −1 is a square modulo p. Ainsi, by Hensel's lemma, 2, −2 or −1 is a square in {style d'affichage mathbb {Q} _{p}} .

An element that is an nth power everywhere locally but not globally 16 is not an 8th power in {style d'affichage mathbb {Q} ({sqrt {7}})} although it is an 8th power locally everywhere (c'est à dire. dans {style d'affichage mathbb {Q} _{p}({sqrt {7}})} for all p). This follows from the above and the equality {style d'affichage mathbb {Q} _{2}({sqrt {7}})=mathbb {Q} _{2}({sqrt {-1}})} .

A consequence of Wang's counter-example Wang's counterexample has the following interesting consequence showing that one cannot always find a cyclic Galois extension of a given degree of a number field in which finitely many given prime places split in a specified way: There exists no cyclic degree 8 extension {displaystyle K/mathbb {Q} } in which the prime 2 is totally inert (c'est à dire., tel que {style d'affichage K_{2}/mathbb {Q} _{2}} is unramified of degree 8).

Special fields For any {displaystyle sgeq 2} laisser {displaystyle eta _{s}:=exp gauche({frac {2pi je}{2^{s}}}droit)+exp gauche(-{frac {2pi je}{2^{s}}}droit)=2cos left({frac {2pi }{2^{s}}}droit).} Note that the {style d'affichage 2 ^{s}} th cyclotomic field is {style d'affichage mathbb {Q} _{2^{s}}=mathbb {Q} (je,eta _{s}).} A field is called s-special if it contains {displaystyle eta _{s}} , but neither {style d'affichage i} , {displaystyle eta _{s+1}} nor {displaystyle ieta _{s+1}} .

Statement of the theorem Consider a number field K and a natural number n. Let S be a finite (possibly empty) set of primes of K and put {style d'affichage K(n,S):={xin Kmid xin K_{mathfrak {p}}^{n}mathrm { pour tous } {mathfrak {p}}not in S}.} The Grunwald–Wang theorem says that {style d'affichage K(n,S)=K^{n}} unless we are in the special case which occurs when the following two conditions both hold: {style d'affichage K} is s-special with an {style d'affichage s} tel que {style d'affichage 2 ^{s+1}} divides n. {style d'affichage S} contains the special set {style d'affichage S_{0}} consisting of those (necessarily 2-adic) nombres premiers {style d'affichage {mathfrak {p}}} tel que {style d'affichage K_{mathfrak {p}}} is s-special.

In the special case the failure of the Hasse principle is finite of order 2: the kernel of {displaystyle K^{fois }/K^{times n}to prod _{{mathfrak {p}}not in S}K_{mathfrak {p}}^{fois }/K_{mathfrak {p}}^{times n}} is Z/2Z, generated by the element ηn s+1.

Explanation of Wang's counter-example The field of rational numbers {displaystyle K=mathbb {Q} } is 2-special since it contains {displaystyle eta _{2}=0} , but neither {style d'affichage i} , {displaystyle eta _{3}={sqrt {2}}} nor {displaystyle ieta _{3}={sqrt {-2}}} . The special set is {style d'affichage S_{0}={2}} . Ainsi, the special case in the Grunwald–Wang theorem occurs when n is divisible by 8, and S contains 2. This explains Wang's counter-example and shows that it is minimal. It is also seen that an element in {style d'affichage mathbb {Q} } is an nth power if it is a p-adic nth power for all p.

The field {displaystyle K=mathbb {Q} ({sqrt {7}})} is 2-special as well, but with {style d'affichage S_{0}= ensemble vide } . This explains the other counter-example above.[1] See also The Hasse norm theorem states that for cyclic extensions an element is a norm if it is a norm everywhere locally. Notes ^ See Chapter X of Artin–Tate. References Artin, Emil; Tate, John (1990), Class field theory, ISBN 978-0-8218-4426-7, M 0223335 Grunwald, Guillaume (1933), "Ein allgemeiner Existenzsatz für algebraische Zahlkörper", Revue de mathématiques pures et appliquées, 169: 103–107 Roquette, Pierre (2005), The Brauer-Hasse-Noether theorem in historical perspective (PDF), Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathematics and Natural Sciences Section of Heidelberg Academy of Sciences], volume. 15, Berlin, New York: Springer Verlag, ISBN 978-3-540-23005-2 Wang, Shianghaw (1948), "A counter-example to Grunwald's theorem", Annales de Mathématiques, Deuxième série, 49: 1008–1009, est ce que je:10.2307/1969410, ISSN 0003-486X, JSTOR 1969410, M 0026992 Wang, Shianghaw (1950), "On Grunwald's theorem", Annales de Mathématiques, Deuxième série, 51: 471–484, est ce que je:10.2307/1969335, ISSN 0003-486X, JSTOR 1969335, M 0033801 Whaples, George (1942), "Non-analytic class field theory and Grünwald's theorem", Journal mathématique de Duke, 9 (3): 455–473, est ce que je:10.1215/s0012-7094-42-00935-9, ISSN 0012-7094, M 0007010 Catégories: Théorie des corps de classes Théorèmes en théorie algébrique des nombres