Conductor of a number field

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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.