27 Functions of Number Theory Multiplicative Number Theory 27.8 Dirichlet Characters 27.10 Periodic Number-Theoretic Functions

§27.9 Quadratic Characters

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Defines:: $(\NVar{n}|\NVar{P})$ : Jacobi symbol and $(\NVar{n}|\NVar{p})$ : Legendre symbol
Keywords:: Dirichlet character, Jacobi symbol, Legendre symbol, characters, number theory, number-theoretic functions, prime numbers, quadratic Jacobi symbol, quadratic Legendre symbol, quadratic characters, quadratic reciprocity law
Notes:: See Apostol (1976, Chapter 9).
Referenced by:: §27.1, §27.1, Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/27.9
See also:: Annotations for Ch.27

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

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

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Symbols:: $(\NVar{n}|\NVar{p})$ : Legendre symbol and : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §27.9 and Ch.27

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

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Symbols:: $(\NVar{n}|\NVar{p})$ : Legendre symbol and : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §27.9 and Ch.27

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

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

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Symbols:: $(\NVar{n}|\NVar{p})$ : Legendre symbol, : odd prime and : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §27.9 and Ch.27

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

27.8 Dirichlet Characters 27.10 Periodic Number-Theoretic Functions

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