If there exists a Rational Integer such that, when , , and are Positive Integers,

then is the -adic reside of , i.e., is an -adic residue of Iff is solvable for . Reciprocity theorems relate statements of the form `` is an -adic residue of '' with reciprocal statements of the form `` is an -adic residue of .''

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.

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1999-05-25