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The Kronecker symbol is a generalization of the Jacobi symbol to all integers.
Let be an integer, with prime factorization
, where is a unit and the are primes. Let be an integer. The Kronecker symbol
is defined to be
For odd , the number
is simply the usual Legendre symbol. This leaves the case when . We define
by
Since it extends the Jacobi symbol, the quantity
is simply 1 when . When , we define it by
These extensions suffice to define the Kronecker symbol for all integer values .
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