Artin's Reciprocity Theorem

A general Reciprocity Theorem for all orders. If $R$ is a Number Field and $R'$ 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 $c$, the kernel of this Surjection contains each Principal fractional Ideal generated by an element congruent to 1 mod $c$.

See also Langlands Program