Gauss reciprocity law

A relation connecting the values of the Legendre symbols (cf. Legendre symbol) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. Quadratic reciprocity law). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation

$$\left(\frac pq\right)\left(\frac qp\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$

there are two more additions to this law, viz.:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}\quad\text{and}\quad\left(\frac2p\right)=(-1)^{(p^2-1)/8}.$$

The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [1]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime.

Attempts to establish the reciprocity law for cubic and biquadratic residues led Gauss to introduce the ring of complex integers.

References

[1]	C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)
[2]	I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)
[3]	H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)

Comments

Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of algebraic number theory and class field theory. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law.