Kronecker symbol

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Note: You might be looking for the Kronecker delta.

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

\left(\frac{a}{2}\right) = \begin{cases} 0 & \mbox{ if }a\mbox{ is even}\\ 1 & \mbox{ if }a\mbox{ is odd and }a \equiv 1\mbox{ or }a \equiv 7 \pmod{8} \\ -1 & \mbox{ if }a\mbox{ is odd and }a \equiv 3\mbox{ or }a \equiv 5 \pmod{8} \end{cases}

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

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

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

[edit] See also

This article incorporates material from Kronecker symbol on PlanetMath, which is licensed under the GFDL.

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