Kronecker symbol

:"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 $p\_i$ are primes. Let $a\; geq\; 0$ be an integer. The Kronecker symbol $left(frac\{a\}\{n\}\; ight)$ is defined to be

:$left(frac\{a\}\{n\}\; ight)\; =\; left(frac\{a\}\{u\}\; ight)\; prod\_\{i=1\}^k\; left(frac\{a\}\{p\_i\}\; ight)^\{e\_i\}.$

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

:

Since it extends the Jacobi symbol, the quantity $left(frac\{a\}\{u\}\; ight)$ 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 $n.$

ee also

* Jacobi symbol
* Legendre symbol

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