Reciprocity Theorem

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.

REFERENCES:

Lemmermeyer, F. Reciprocity Laws: Their Evolution from Euler to Artin. Berlin: Springer-Verlag, 2000.

Lemmermeyer, F. "Bibliography on Reciprocity Laws." https://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.