# Conductor of a number field

In algebraic number theory, the **conductor** or **relative conductor** of an extension of algebraic number fields is a modulus which determines the splitting of prime ideals. If no extension is specified, then the **absolute conductor** refers to a number field regarded as an extension of the field of rational numbers. There need not be a conductor for an extension: indeed, class field theory shows that one exists precisely when the extension is abelian.

There is a simple description of the absolute conductor. By the Kronecker-Weber theorem, every abelian extension of **Q** lies in some cyclotomic field, that is, an extension by roots of unity. The absolute conductor of an abelian number field *F* is then the smallest integer *f* such that *F* is a subfield of the field of *f*-th roots of unity.

A quadratic field is always abelian. In this case the conductor is equal to the field discriminant.

For a general extension *F*/*K*, the conductor is a modulus of *K*.