The **Jacobi symbol** allows any odd positive integer (not just odd positive primes) to be in the bottom of the Legendre symbols. It is given by $(\frac{a}{n})$ where $n=p_1^{e_1}p_2^{e_2}...p_r^{e_r}$, then

Where $(\frac{a}{p})$ is a Legendre symbol, with p-prime and odd. It is a convention to set $(\frac{a}{1})=1$.

It follows the same properties as in Proposition 4.4a,b,c from the text and has the additional property that $(\frac{a}{nm})=(\frac{a}{n})(\frac{a}{m})$.

Note that the Jacobi symbol is a generalization of the Legendre Symbol. There is also a generalization of the Jacobi symbol which allows any integer to be in the "bottom" of the symbol, so it drops the "it must be odd" condition which Jacobi has. It is called the **Kronecker symbol**.

Here is a link that calculates Jacobi symbols - this could help double check some homework questions!

http://www.math.fau.edu/richman/jacobi.htm