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Kronecker symbol
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 .
Related:
JacobiSymbol, LegendreSymbol
Synonym:
Kronecker-Jacobi symbol
Type of Math Object:
Definition
Major Section:
Reference
Mathematics Subject Classification
11A07 Congruences; primitive roots; residue systems11A15 Power residues, reciprocity
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