Matches in DBpedia 2014 for { ?s ?p 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.. }
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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.".