Quadratic reciprocity law

The relation

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

connecting the Legendre symbols (cf. Legendre symbol)

$$\left(\frac pq\right)\quad\text{and}\quad\left(\frac qp\right)$$

for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, namely:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}$$

and

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

C.F. Gauss gave the first complete proof of the quadratic reciprocity law, which for this reason is also called the Gauss reciprocity law.

It immediately follows from this law that for a given square-free number $d$, the primes $p$ for which $d$ is a quadratic residue modulo $p$ ly in certain arithmetic progressions with common difference $2|d|$ or $4|d|$. The number of these progressions is $\phi(2|d|)/2$ or $\phi(4|d|)/2$, where $\phi(n)$ is the Euler function. The quadratic reciprocity law makes it possible to establish factorization laws in quadratic extensions $\mathbf Q(\sqrt d)$ of the field of rational numbers, since the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a prime number that does not divide $d$ depends on whether or not $x^2-d$ is reducible modulo $p$.

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)

Comments

See also Quadratic residue; Dirichlet character.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII