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- Cyclotomic_unit abstract "In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζan − 1) for ζn an nth root of unity and 0 < a < n. Note that if n is the power of a prime ζan − 1 itself is not a unit; however the numbers (ζan − 1)/(ζn − 1) for (a, n) = 1, and ±ζan generate the group of cyclotomic units in this case (n power of a prime). The cyclotomic units form a subgroup of finite index in the group of units of a cyclotomic field. The index of this subgroup of real cyclotomic units (those cyclotomic units in the maximal real subfield) within the full unit group is equal to the class number of the maximal real subfield of the cyclotomic field.Note also that if n is a composite number, the subgroup of cyclotomic units generated by (ζan − 1)/(ζn − 1)with (a, n) = 1 is not of finite index in general.The cyclotomic units satisfy distribution relations. Let a be a rational number prime to p and let ga denote exp(2πia)−1. Then for a≠ 0 we have .Using these distribution relations and the symmetry relation ζan − 1 = -ζan (ζ-an − 1) a basis Bn of the cyclotomic units can be constructed with the property that Bd ⊆ Bn for d | n.".
- Cyclotomic_unit wikiPageID "5991396".
- Cyclotomic_unit wikiPageRevisionID "595024007".
- Cyclotomic_unit b "n".
- Cyclotomic_unit hasPhotoCollection Cyclotomic_unit.
- Cyclotomic_unit p "-a".
- Cyclotomic_unit p "a".
- Cyclotomic_unit subject Category:Algebraic_number_theory.
- Cyclotomic_unit subject Category:Cyclotomic_fields.
- Cyclotomic_unit type CyclotomicFields.
- Cyclotomic_unit type Field108569998.
- Cyclotomic_unit type GeographicalArea108574314.
- Cyclotomic_unit type Location100027167.
- Cyclotomic_unit type Object100002684.
- Cyclotomic_unit type PhysicalEntity100001930.
- Cyclotomic_unit type Region108630985.
- Cyclotomic_unit type Tract108673395.
- Cyclotomic_unit type YagoGeoEntity.
- Cyclotomic_unit type YagoLegalActorGeo.
- Cyclotomic_unit type YagoPermanentlyLocatedEntity.
- Cyclotomic_unit comment "In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζan − 1) for ζn an nth root of unity and 0 < a < n. Note that if n is the power of a prime ζan − 1 itself is not a unit; however the numbers (ζan − 1)/(ζn − 1) for (a, n) = 1, and ±ζan generate the group of cyclotomic units in this case (n power of a prime). The cyclotomic units form a subgroup of finite index in the group of units of a cyclotomic field.".
- Cyclotomic_unit label "Cyclotomic unit".
- Cyclotomic_unit sameAs m.0fjj9v.
- Cyclotomic_unit sameAs Q5199069.
- Cyclotomic_unit sameAs Q5199069.
- Cyclotomic_unit sameAs Cyclotomic_unit.
- Cyclotomic_unit wasDerivedFrom Cyclotomic_unit?oldid=595024007.
- Cyclotomic_unit isPrimaryTopicOf Cyclotomic_unit.