The Kronecker symbol is a generalization of the Jacobi symbol to all integers.

Let $n$ be an integer, with prime factorization $u \cdot {p_1}^{e_1} \cdots {p_k}^{e_k}$ , where $u$ is a unit and the $p_i$ are primes. Let $a \geq 0$ be an integer. The Kronecker symbol $\left(\frac{a}{n}\right)$ is defined to be $$ \left(\frac{a}{n}\right) = \left(\frac{a}{u}\right) \prod_{i=1}^k \left(\frac{a}{p_i}\right)^{e_i}$$

For odd $p_i$ , the number $\left(\frac{a}{p_i}\right)$ is simply the usual Legendre symbol. This leaves the case when $p_i=2$ . We define $\left(\frac{a}{2}\right)$ by

Since it extends the Jacobi symbol, the quantity $\left(\frac{a}{u}\right)$ is simply 1 when $u=1$ . When $u=-1$ , we define it by

These extensions suffice to define the Kronecker symbol for all integer values $n$ .