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Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in.

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  • Der tschebotarjowsche Dichtigkeitssatz (je nach Transkription auch Dichtigkeitssatz von Chebotarëv oder Tschebotareff) ist eine Verallgemeinerung des Satzes von Dirichlet über Primzahlen in arithmetischen Progressionen auf Galoiserweiterungen von Zahlkörpern. Im Falle einer abelschen Erweiterung von erhält man daraus den Satz zurück, dass die Menge der Primzahlen der Form , hat. In seiner allgemeinen Form folgt daraus insbesondere der 1880 von Kronecker bewiesene Satz, dass genau der Primzahlen in einer gegebenen Galoiserweiterung von vom Grad sind. Der Satz wurde von Nikolai Grigorjewitsch Tschebotarjow im Jahr 1922 gefunden und 1923 erstmals auf russisch, 1925 auf deutsch veröffentlicht. (de)
  • Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in. A special case that is easier to state says that if K is an algebraic number field which is a Galois extension of of degree n, then the prime numbers that completely split in K have density 1/n among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which is a representative of a well-defined conjugacy class in the Galois group Gal(K/Q). Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with k elements occurs with frequency asymptotic to k/n. (en)
  • En théorie algébrique des nombres, le théorème de Tchebotariov, dû à Nikolai Tchebotariov et habituellement écrit théorème de Chebotarev, précise le théorème de la progression arithmétique de Dirichlet sur l'infinitude des nombres premiers en progression arithmétique : il affirme que, si a, q ≥ 1 sont deux entiers premiers entre eux, la densité naturelle de l'ensemble des nombres premiers congrus à a modulo q vaut 1/φ(q). (fr)
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  • En théorie algébrique des nombres, le théorème de Tchebotariov, dû à Nikolai Tchebotariov et habituellement écrit théorème de Chebotarev, précise le théorème de la progression arithmétique de Dirichlet sur l'infinitude des nombres premiers en progression arithmétique : il affirme que, si a, q ≥ 1 sont deux entiers premiers entre eux, la densité naturelle de l'ensemble des nombres premiers congrus à a modulo q vaut 1/φ(q). (fr)
  • Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in. (en)
  • Der tschebotarjowsche Dichtigkeitssatz (je nach Transkription auch Dichtigkeitssatz von Chebotarëv oder Tschebotareff) ist eine Verallgemeinerung des Satzes von Dirichlet über Primzahlen in arithmetischen Progressionen auf Galoiserweiterungen von Zahlkörpern. Im Falle einer abelschen Erweiterung von erhält man daraus den Satz zurück, dass die Menge der Primzahlen der Form , hat. In seiner allgemeinen Form folgt daraus insbesondere der 1880 von Kronecker bewiesene Satz, dass genau der Primzahlen in einer gegebenen Galoiserweiterung von vom Grad sind. (de)
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  • Tschebotarjowscher Dichtigkeitssatz (de)
  • Chebotarev's density theorem (en)
  • Théorème de densité de Tchebotariov (fr)
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