Conductor of a number field: Difference between revisions
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In [[algebraic number theory]], the '''conductor''' or '''relative conductor''' of an [[field extension|extension]] of [[algebraic number field]]s is a [[modulus (algebraic number theory)|modulus]] which determines the splitting of [[prime ideal]]s. If no extension is specified, then the '''absolute conductor''' refers to a number field regarded as an extension of the field of [[rational number]]s. There need not be a conductor for an extension: indeed, [[class field theory]] shows that one exists precisely when the extension is abelian. | In [[algebraic number theory]], the '''conductor''' or '''relative conductor''' of an [[field extension|extension]] of [[algebraic number field]]s is a [[modulus (algebraic number theory)|modulus]] which determines the splitting of [[prime ideal]]s. If no extension is specified, then the '''absolute conductor''' refers to a number field regarded as an extension of the field of [[rational number]]s. There need not be a conductor for an extension: indeed, [[class field theory]] shows that one exists precisely when the extension is abelian. | ||
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A [[quadratic field]] is always abelian. In this case the conductor is equal to the [[field discriminant]]. | 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 (algebraic number theory)|modulus]] of ''K''. | For a general extension ''F''/''K'', the conductor is a [[modulus (algebraic number theory)|modulus]] of ''K''.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 07:00, 1 August 2024
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.