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Concept
Kronecker symbol
Summary
In number theory, the Kronecker symbol, written as or , is a generalization of the Jacobi symbol to all integers . It was introduced by .
Let be a non-zero integer, with prime factorization
where is a unit (i.e., ), and the are primes. Let be an integer. The Kronecker symbol is defined by
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 when . When , we define it by
Finally, we put
These extensions suffice to define the Kronecker symbol for all integer values .
Some authors only define the Kronecker symbol for more restricted values; for example, congruent to and .
The following is a table of values of Kronecker symbol with 1 ≤ n, k ≤ 30.
The Kronecker symbol shares many basic properties of the Jacobi symbol, under certain restrictions:
if , otherwise .
unless , one of is zero and the other one is negative.
unless , one of is zero and the other one has odd part (definition below) congruent to .
For , we have whenever If additionally have the same sign, the same also holds for .
For , , we have whenever
On the other hand, the Kronecker symbol does not have the same connection to quadratic residues as the Jacobi symbol. In particular, the Kronecker symbol for even can take values independently on whether is a quadratic residue or nonresidue modulo .
The Kronecker symbol also satisfies the following versions of quadratic reciprocity law.
For any nonzero integer , let denote its odd part: where is odd (for , we put ). Then the following symmetric version of quadratic reciprocity holds for every pair of integers such that :
where the sign is equal to if or and is equal to if and .
There is also equivalent non-symmetric version of quadratic reciprocity that holds for every pair of relatively prime integers :
For any integer let . Then we have another equivalent non-symmetric version that states
for every pair of integers (not necessarily relatively prime).
The supplementary laws generalize to the Kronecker symbol as well.
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