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Difference between revisions of "Jacobi symbol"

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(Comment: Extension to Legendre–Jacobi–Kronecker symbol, cite Cohen (1993))
 
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$$\left(\frac aP\right)$$
 
$$\left(\frac aP\right)$$
  
A function defined for all integers $a$ coprime to a given odd integer $P>1$ as follows: Let $P=p_1\ldots p_r$ be an expansion of $P$ into prime factors (not necessarily different), then
+
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
  
$$\left(\frac aP\right)=\left(\frac{a}{p_1}\right)\ldots\left(\frac{a}{p_r}\right),$$
+
$$\left(\frac aP\right)=\left(\frac{a}{p_1}\right)\dotsm\left(\frac{a}{p_r}\right),$$
  
 
where
 
where
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<table>
 
<table>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Zagier,  "Zetafunktionen und quadratische Körper" , Springer  (1981)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Zagier,  "Zetafunktionen und quadratische Körper" , Springer  (1981)</TD></TR>
<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>
+
<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>
 
</table>
 
</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=35652
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article