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'' <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541201.png" /></td> </tr></table>
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$$\left(\frac aP\right)$$
  
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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A function defined for all integers $a$ coprime to a given odd integer $P>1$ as follows: Let $P=p_1\dotsm p_r$ be an expansion of $P$ into prime factors (not necessarily different), then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541206.png" /></td> </tr></table>
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$$\left(\frac aP\right)=\left(\frac{a}{p_1}\right)\dotsm\left(\frac{a}{p_r}\right),$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541207.png" /></td> </tr></table>
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$$\left(\frac{a}{p_i}\right)$$
  
 
is the [[Legendre symbol|Legendre symbol]].
 
is the [[Legendre symbol|Legendre symbol]].
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The Jacobi symbol is a generalization of the Legendre symbol and has similar properties. In particular, the reciprocity law:
 
The Jacobi symbol is a generalization of the Legendre symbol and has similar properties. In particular, the reciprocity law:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j0541208.png" /></td> </tr></table>
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$$\left(\frac PQ\right)\left(\frac QP\right)=(-1)^{(P-1)/2\cdot(Q-1)/2}$$
  
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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holds, where $P$ and $Q$ are positive odd coprime numbers, and the supplementary formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054120/j05412011.png" /></td> </tr></table>
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$$\left(\frac{-1}{P}\right)=(-1)^{(P-1)/2},\quad\left(\frac 2P\right)=(-1)^{(P^2-1)/8}$$
  
 
are true.
 
are true.
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====Comments====
 
====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]]]).
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Considered as a function on $(\mathbf Z/p\mathbf Z)^*$, 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]]]).
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There is a further extension to the case of arbitrary $P$, the Kronecker, or [[Legendre–Jacobi–Kronecker symbol]]
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Zagier,  "Zetafunktionen und quadratische Körper" , Springer  (1981)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Zagier,  "Zetafunktionen und quadratische Körper" , Springer  (1981)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  Henri Cohen, ''A Course in Computational Algebraic Number Theory'', Graduate Texts in Mathematics '''138''' Springer (1993) {{ISBN|3-540-55640-0}}</TD></TR>
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</table>

Latest revision as of 17:41, 11 November 2023

2020 Mathematics Subject Classification: Primary: 11A15 [MSN][ZBL]

$$\left(\frac aP\right)$$

A function defined for all integers $a$ coprime to a given odd integer $P>1$ as follows: Let $P=p_1\dotsm p_r$ be an expansion of $P$ into prime factors (not necessarily different), then

$$\left(\frac aP\right)=\left(\frac{a}{p_1}\right)\dotsm\left(\frac{a}{p_r}\right),$$

where

$$\left(\frac{a}{p_i}\right)$$

is the Legendre symbol.

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

$$\left(\frac PQ\right)\left(\frac QP\right)=(-1)^{(P-1)/2\cdot(Q-1)/2}$$

holds, where $P$ and $Q$ are positive odd coprime numbers, and the supplementary formulas

$$\left(\frac{-1}{P}\right)=(-1)^{(P-1)/2},\quad\left(\frac 2P\right)=(-1)^{(P^2-1)/8}$$

are true.

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

References

[1] C.G.J. Jacobi, "Gesammelte Werke" , 1–7 , Reimer (1881–1891)
[2] P.G.L. Dirichlet, "Vorlesungen über Zahlentheorie" , Vieweg (1894)
[3] P. Bachmann, "Niedere Zahlentheorie" , 1–2 , Teubner (1902–1910)


Comments

Considered as a function on $(\mathbf Z/p\mathbf Z)^*$, 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 (see [a1]).

There is a further extension to the case of arbitrary $P$, the Kronecker, or Legendre–Jacobi–Kronecker symbol

References

[a1] D.B. Zagier, "Zetafunktionen und quadratische Körper" , Springer (1981)
[a1] Henri Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138 Springer (1993) ISBN 3-540-55640-0
How to Cite This Entry:
Jacobi symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_symbol&oldid=11678
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article