§ 27.9. Quadratic Characters

Notes:: See Apostol (1976, Chapter 9).
Referenced by:: Tab.27.1.1
Permalink:: http://dlmf.nist.gov/27.9

For an odd prime $p$ , the Legendre symbol $(n|p)$ is defined as follows. If $p$ divides $n$ , then the value of $(n|p)$ is 0. If $p$ does not divide $n$ , then $(n|p)$ has the value 1 when the quadratic congruence $x^{2}\equiv n\;\;(\textrm{mod}p)$ has a solution, and the value −1 when this congruence has no solution. The Legendre symbol $(n|p)$ , 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 $(-1|p)=(-1)^{{(p-1)/2}},$

Defines:: $(n|p)$ : Legendre symbol
Symbols:: $p$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E1
Encodings:: TeX, pMathML, png

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

Defines:: $(n|p)$ : Legendre symbol
Symbols:: $p$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E2
Encodings:: TeX, pMathML, png

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

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

Defines:: $(n|p)$ : Legendre symbol, $p$ : odd prime and $q$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E3
Encodings:: TeX, pMathML, png

If an odd integer $P$ has prime factorization $P=\prod _{{r=1}}^{{\nu\!\left(n\right)}}p^{{a_{r}}}_{r}$ , then the Jacobi symbol $(n|P)$ is defined by $(n|P)=\prod _{{r=1}}^{{\nu\!\left(n\right)}}{(n|p_{r})^{{a_{r}}}}$ , with $(n|1)=1$ . The Jacobi symbol $(n|P)$ 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 $P,Q$ .