Class field theory makes sense for arbitrary number fields, but for simplicity in this section and because it is all that is needed for our application to the BSD conjecture, we assume henceforth that is a totally imaginary number field, i.e., one with no real embeddings.
Let be a finite abelian extension of number fields, and let be any unramified prime ideal in . Let be an prime of over and consider the extension of the finite field . There is an element that acts via th powering on , where . A basic fact one proves in algebraic number theory is that there is an element that acts as on ; moreover, replacing by a different ideal over just changes by conjugation. Since is abelian it follows that is uniquely determined by . The association is called the Artin reciprocity map.
Let be an integral
ideal divisible by all primes of that ramify in , and let
be the group of fractional ideals that are coprime to .
Then the reciprocity map extends to a map
If is the conductor of then Artin reciprocity induces
a group homomorphism
In particular, since the Hilbert class field is unramified over , we have:
William 2007-05-25