If there exists a Rational Integer such that, when , , and are Positive Integers,
The first case to be considered was (the Quadratic Reciprocity Theorem), of which Gauß gave the first correct proof. Gauss also solved the case (Cubic Reciprocity Theorem) using Integers of the form , where is a root of and , are rational Integers. Gauß stated the case (Quartic Reciprocity Theorem) using the Gaussian Integers.
Proof of -adic reciprocity for Prime was given by Eisenstein in 1844-50 and by Kummer in 1850-61. In the 1920s, Artin formulated Artin's Reciprocity Theorem, a general reciprocity law for all orders.
See also Artin Reciprocity, Cubic Reciprocity Theorem, Langlands Reciprocity, Quadratic Reciprocity Theorem, Quartic Reciprocity Theorem, Rook Reciprocity Theorem