If there exists a rational integer positive integers ,
then iff of
the form "
The first case to be considered was quadratic
reciprocity theorem ), of which Gauss gave the first correct proof. Gauss also
solved the case cubic reciprocity theorem ) using integers of the form integers . Gauss stated
the case biquadratic reciprocity theorem )
using the Gaussian integers .
Proof of prime 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
Explore with Wolfram|Alpha
References Lemmermeyer, F. Reciprocity Laws: Their Evolution from Euler to Artin. 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. 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 https://mathworld.wolfram.com/ReciprocityTheorem.html
Subject classifications
such that, when 
, 
, and 
are positive integers,
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 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.
-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.
