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Chebotarev density theorem

From Encyclopedia of Mathematics
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Let be a normal (finite-degree) extension of algebraic number fields with Galois group . Pick a prime ideal of and let be the prime ideal of under it, i.e. , where is the ring of integers of . There is a unique element

of such that for integral. Here, , the norm of , is the number of elements of the residue field . This is the Frobenius automorphism (or Frobenius symbol) associated to .

If is unramified in , define as the conjugacy class of in , where is any prime ideal above . This conjugacy class depends only on .

The weak form of the Chebotarev density theorem says that if is an arbitrary conjugacy class in , then the set

is infinite and has Dirichlet density , where .

The stonger form specifies in addition that is regular (see Dirichlet density) and that

with the number of prime ideals in with norm .

References

[a1] W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.3 (Edition: Second)
[a2] N.G. Chebotarev, "Determination of the density of the set of primes corresponding to a given class of permutations" Izv. Akad. Nauk. , 17 (1923) pp. 205–230; 231–250 (In Russian)
[a3] N.G. Chebotarev, "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen welche zu einer gegebenen Substitutionsklasse gehören" Math. Ann. , 95 (1926) pp. 191–228
How to Cite This Entry:
Chebotarev density theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebotarev_density_theorem&oldid=35122
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article