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Let be a non-zero integer, with prime factorization
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 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
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 integers (not necessarily relatively prime):
For any integer let . Then we have another equivalent non-symmetric version that states
for every pair of integers .
The supplementary laws generalize to the Kronecker symbol as well. These laws follow easily from each version of quadratic reciprocity law stated above (unlike with Legendre and Jacobi symbol where both the main law and the supplementary laws are needed to fully describe the quadratic reciprocity).
For any integer we have
and for any odd integer it's
Connection to Dirichlet characters
If and , the map is a real Dirichlet character of modulus Conversely, every real Dirichlet character can be written in this form with (for it's ).
In particular, primitive real Dirichlet characters are in a 1–1 correspondence with quadratic fields , where is a nonzero square-free integer (we can include the case to represent the principal character, even though it is not a proper quadratic field). The character can be recovered from the field as the Artin symbol : that is, for a positive prime , the value of depends on the behaviour of the ideal in the ring of integers :
Then equals the Kronecker symbol , where
is the discriminant of . The conductor of is .
Similarly, if , the map is a real Dirichlet character of modulus However, not all real characters can be represented in this way, for example the character cannot be written as for any . By the law of quadratic reciprocity, we have . A character can be represented as if and only if its odd part , in which case we can take .
- Kronecker, L. (1885), "Zur Theorie der elliptischen Funktionen", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 761–784
- Montgomery, Hugh L; Vaughan, Robert C. (2007). Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics 97. Cambridge University Press . ISBN 0-521-84903-9. Zbl 1142.11001.