Reciprocity Theorem

If there exists a rational integer x such that, when n, p, and q are positive integers,

 x^n=q (mod p),

then q is the n-adic residue of p, i.e., q is an n-adic residue of p iff x^n=q (mod p) 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+brho, where rho is a root of x^2+x+1=0 and a, b are rational integers. Gauss stated the case n=4 (biquadratic 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'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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Lemmermeyer, F. Reciprocity Laws: Their Evolution from Euler to Artin. Berlin: Springer-Verlag, 2000.Lemmermeyer, F. "Bibliography on Reciprocity Laws.", 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.

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Reciprocity Theorem

Cite this as:

Weisstein, Eric W. "Reciprocity Theorem." From MathWorld--A Wolfram Web Resource.

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