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
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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=35122