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