Artin's Reciprocity Theorem

A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. 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 .