Take a number field and an Abelian extension, then form a prime divisor that is divided by all ramified primes of the extension . Now define a map from the fractional ideals relatively prime to to the Galois group of that sends an ideal to . This map is called the Artin map. Its importance lies in the kernel, which Artin's reciprocity theorem states contains all fractional ideals that are only composed of primes that split completely in the extension .
This is the reason that it is a reciprocity law. The inertia degree of a prime can now be computed since the smallest exponent for which belongs to this kernel, which is exactly the inertia degree, is now known. Now because is unramified and is Galois, , with the inertia degree and the number of factors into which splits when it is extended to . So it is completely known how behaves when it is extended to .
This is completely analogous to quadratic reciprocity because it also determines when an unramified prime splits (, ) or is inert (, ). Of course, quadratic reciprocity is much simpler since there are only two possibilities in this case.
In the special case of the Hilbert class field, this kernel coincides with the ordinary class group of the extension.
a number field and 
an Abelian extension, then form a prime divisor that is divided by all ramified primes of the extension 
. Now define a map 
from the fractional ideals relatively prime to 
to the Galois group of 
that sends an ideal 
to 
. This map is called the Artin map. Its importance lies in the kernel, which Artin's reciprocity theorem states contains all fractional ideals that are only composed of primes that split completely in the extension 
. 
for which 
belongs to this kernel, which is exactly the inertia degree, is now known. Now because 
is unramified and 
is Galois,
, with 
the inertia degree and 
the number of factors into which 
splits when it is extended to 
. So it is completely known how 
behaves when it is extended to 
. 
splits (
,
) or is inert (
,
). Of course, quadratic reciprocity is much simpler since there are only two possibilities in this case.