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

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 $n$.