Artin's Reciprocity Theorem

Artin's reciprocity theorem is a general reciprocity theorem that subsumes the previously known reciprocity theorems. If is a number field and a finite integral extension, then there is a surjection from the group of fractional ideals prime to the discriminant, given by the Artin symbol. For some cycle , the kernel of this surjection contains each principal fractional ideal generated by an element congruent to 1 mod .