View source for Jacobi symbol

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A function defined for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541202.png" /> coprime to a given odd integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541203.png" /> as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541204.png" /> be an expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541205.png" /> into prime factors (not necessarily different), then

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where

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is the [[Legendre symbol|Legendre symbol]].

The Jacobi symbol is a generalization of the Legendre symbol and has similar properties. In particular, the reciprocity law:

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holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j05412010.png" /> are positive odd coprime numbers, and the supplementary formulas

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are true.

The Jacobi symbol was introduced by C.G.J. Jacobi (1837).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G.J. Jacobi,   "Gesammelte Werke" , '''1–7''' , Reimer  (1881–1891)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.G.L. Dirichlet,   "Vorlesungen über Zahlentheorie" , Vieweg  (1894)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Bachmann,   "Niedere Zahlentheorie" , '''1–2''' , Teubner  (1902–1910)</TD></TR></table>



====Comments====
Considered as a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j05412012.png" />, the Jacobi symbol is an example of a real character. This real character plays an important role in the decomposition of rational primes in a [[Quadratic field|quadratic field]] (see [[#References|[a1]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Zagier,   "Zetafunktionen und quadratische Körper" , Springer  (1981)</TD></TR></table>