If there exists a rational integer such that, when , , and are positive integers ,

then
is the -adic
residue 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
(biquadratic 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's Reciprocity Theorem ,

Biquadratic Reciprocity Theorem ,

Class Field Theory ,

Class
Number ,

Cubic Reciprocity Theorem ,

Langlands Program ,

Langlands
Reciprocity ,

Octic Reciprocity Theorem ,

Quadratic Reciprocity Theorem ,

Rook Reciprocity Theorem
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References Lemmermeyer, F. Reciprocity Laws: Their Evolution from Euler to Artin. Berlin: Springer-Verlag, 2000. Lemmermeyer,
F. "Bibliography on Reciprocity Laws." http://www.rzuser.uni-heidelberg.de/~hb3/recbib.html . Nagell,
T. "Power Residues. Binomial Congruences." §34 in Introduction
to Number Theory. New York: Wiley, pp. 115-120, 1951. Wyman,
B. F. "What Is a Reciprocity Law?" Amer. Math. Monthly 79 ,
571-586, 1972. Referenced on Wolfram|Alpha Reciprocity Theorem
Cite this as:
Weisstein, Eric W. "Reciprocity Theorem."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ReciprocityTheorem.html

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