Chebotarev density theorem
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 |
Chebotarev density theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebotarev_density_theorem&oldid=13409