Jacobi symbol

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 $P=p_1\ldots p_r$ be an expansion of $P$ into prime factors (not necessarily different), then

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

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