§27.9 Quadratic Characters

Notes:: See Apostol (1976, Chapter 9).
Defines:: $\mathop{(n|P)\/}\nolimits$ : Jacobi symbol and $\mathop{(n|p)\/}\nolimits$ : Legendre symbol
Keywords:: Jacobi symbol, Legendre symbol, characters, number-theoretic functions, number theory, prime numbers, quadratic characters, quadratic reciprocity law
Referenced by:: §27.1
Permalink:: http://dlmf.nist.gov/27.9

For an odd prime $p$ , the Legendre symbol $\mathop{(n|p)\/}\nolimits$ is defined as follows. If $p$ divides $n$ , then the value of $\mathop{(n|p)\/}\nolimits$ is 0. If $p$ does not divide $n$ , then $\mathop{(n|p)\/}\nolimits$ has the value 1 when the quadratic congruence $x^{2}\equiv n\;\;(\mathop{{\rm mod}}p)$ has a solution, and the value −1 when this congruence has no solution. The Legendre symbol $\mathop{(n|p)\/}\nolimits$ , as a function of $n$ , is a Dirichlet character (mod $p$ ). It is sometimes written as $(\frac{n}{p})$ . Special values include:

27.9.1 $\mathop{(-1|p)\/}\nolimits=(-1)^{{(p-1)/2}},$

Symbols:: $\mathop{(n|p)\/}\nolimits$ : Legendre symbol and $p$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E1
Encodings:: TeX, pMML, png

27.9.2 $\mathop{(2|p)\/}\nolimits=(-1)^{{(p^{2}-1)/8}}.$

Symbols:: $\mathop{(n|p)\/}\nolimits$ : Legendre symbol and $p$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E2
Encodings:: TeX, pMML, png

If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that

27.9.3 $\mathop{(p|q)\/}\nolimits\mathop{(q|p)\/}\nolimits=(-1)^{{(p-1)(q-1)/4}}.$

Symbols:: $\mathop{(n|p)\/}\nolimits$ : Legendre symbol, $p$ : odd prime and $q$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E3
Encodings:: TeX, pMML, png

If an odd integer $P$ has prime factorization $P=\prod _{{r=1}}^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}p^{{a_{r}}}_{r}$ , then the Jacobi symbol $\mathop{(n|P)\/}\nolimits$ is defined by $\mathop{(n|P)\/}\nolimits=\prod _{{r=1}}^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}{\mathop{(n|p_{r})\/}\nolimits^{{a_{r}}}}$ , with $\mathop{(n|1)\/}\nolimits=1$ . The Jacobi symbol $\mathop{(n|P)\/}\nolimits$ is a Dirichlet character (mod $P$ ). Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$ ; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$ .