In number theory, the Kronecker symbol, written as
, is a generalization of the Jacobi symbol to all integers
It was introduced by Leopold Kronecker (1885, page 770).
be a non-zero integer, with prime factorization where
is simply the usual Legendre symbol.
by Since it extends the Jacobi symbol, the quantity
, 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,
The following is a table of values of Kronecker symbol
The Kronecker symbol shares many basic properties of the Jacobi symbol, under certain restrictions: On the other hand, the Kronecker symbol does not have the same connection to quadratic residues as the Jacobi symbol.
is a quadratic residue or nonresidue modulo
The Kronecker symbol also satisfies the following versions of quadratic reciprocity law.
denote its odd part:
Then the following symmetric version of quadratic reciprocity holds for every pair of integers
There is also equivalent non-symmetric version of quadratic reciprocity that holds for every pair of relatively prime integers
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).
is a real Dirichlet character of modulus
Conversely, every real Dirichlet character can be written in this form with
In particular, primitive real Dirichlet characters
are in a 1–1 correspondence with quadratic fields
is a nonzero square-free integer (we can include the case
to represent the principal character, even though it is not a quadratic field).
can be recovered from the field as the Artin symbol
depends on the behaviour of the ideal
equals the Kronecker symbol
is a real Dirichlet character of modulus
By the law of quadratic reciprocity, we have
This article incorporates material from Kronecker symbol on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.