Kronecker Symbol - Connection To Dirichlet Characters

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.

In particular, primitive real Dirichlet characters are in a 1–1 correspondence with quadratic fields, where m 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 p, 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 F. 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 n. 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 .