Kronecker Symbol - Definition

Definition

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

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

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

Finally, we put

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

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