Reciprocity Theorem

If there exists a Rational Integer $x$ such that, when $n$, $p$, and $q$ are Positive Integers,

$$x^n\equiv q\ \left({{\rm mod\ } {p}}\right),$$

then $q$ is the $n$-adic reside of $p$, i.e., $q$ is an $n$-adic residue of $p$ Iff $x^n\equiv q\ \left({{\rm mod\ } {p}}\right)$ is solvable for $x$. Reciprocity theorems relate statements of the form ``$p$ is an $n$-adic residue of $q$'' with reciprocal statements of the form ``$q$ is an $n$-adic residue of $p$.''

The first case to be considered was $n=2$ (the Quadratic Reciprocity Theorem), of which Gauß gave the first correct proof. Gauss also solved the case $n=3$ (Cubic Reciprocity Theorem) using Integers of the form $a+b\rho$, where $\rho$ is a root of $x^2+x+1=0$ and $a$, $b$ are rational Integers. Gauß stated the case $n=4$ (Quartic Reciprocity Theorem) using the Gaussian Integers.

Proof of $n$-adic reciprocity for Prime $n$ 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