If there exists a
rational integer such that, when , , and are positive integers,
is the -adic
residue of ,
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
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
( biquadratic reciprocity theorem)
using the Gaussian integers.
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
Cubic Reciprocity Theorem
Octic Reciprocity Theorem
Quadratic Reciprocity Theorem
Rook Reciprocity Theorem
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References Lemmermeyer, F. Berlin: Springer-Verlag, 2000. 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 New York: Wiley, pp. 115-120, 1951. 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 --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/ReciprocityTheorem.html