Kronecker symbol

In number theory, the Kronecker symbol, written as \( \left(\frac an\right) \) or (a|n), is a generalization of the Jacobi symbol to all integers n. It was introduced by Leopold Kronecker.
Definition

Let n be a non-zero integer, with prime factorization

\( u \cdot {p_1}^{e_1} \cdots {p_k}^{e_k}, \)

where u is a unit (i.e., u is 1 or −1), and the pi are primes. Let a be an integer. The Kronecker symbol (a|n) is defined by

\( \left(\frac{a}{n}\right) = \left(\frac{a}{u}\right) \prod_{i=1}^k \left(\frac{a}{p_i}\right)^{e_i}. \)

For odd pi, the number (a|pi) is simply the usual Legendre symbol. This leaves the case when pi = 2. We define (a|2) by

\( \left(\frac{a}{2}\right) = \begin{cases} 0 & \mbox{if }a\mbox{ is even,} \\ 1 & \mbox{if } a \equiv \pm1 \pmod{8}, \\ -1 & \mbox{if } a \equiv \pm3 \pmod{8}. \end{cases} \)

Since it extends the Jacobi symbol, the quantity (a|u) is simply 1 when u = 1. When u = −1, we define it by

\( \left(\frac{a}{-1}\right) = \begin{cases} -1 & \mbox{if }a < 0, \\ 1 & \mbox{if } a \ge 0. \end{cases} \)

Finally, we put

\( \left(\frac a0\right)=\begin{cases}1&\text{if }a=\pm1,\\0&\text{otherwise.}\end{cases} \)

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

Unlike the Jacobi symbol, the Kronecker symbol \( (\tfrac{a}{n}) \) is not a Dirichlet character to the modulus n, because \( a \equiv b \pmod{n} does not imply \( (\tfrac{a}{n}) = (\tfrac{b}{n}) \) when n is even.

This article incorporates material from Kronecker symbol on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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