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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 (ζ − 1) for ζ an ''n''th root of unity and 0 < ''a'' < ''n''. Note that if ''n'' is the power of a prime ζ − 1 itself is not a unit; however the numbers (ζ − 1)/(ζ − 1) for (''a'', ''n'') = 1, and ±ζ 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.〔Washington, Theorem 8.2〕 Note also that if ''n'' is a composite number, the subgroup of cyclotomic units generated by (ζ − 1)/(ζ − 1)with (''a'', ''n'') = 1 is not of finite index in general.〔Washington, 8.8, page 150, for ''n'' equal to 55.〕 The cyclotomic units satisfy ''distribution relations''. Let ''a'' be a rational number prime to ''p'' and let ''g''''a'' denote exp(2πi''a'')−1. Then for ''a''≠ 0 we have .〔Lang (1990) p.157〕 Using these distribution relations and the symmetry relation ζ − 1 = -ζ (ζ − 1) a basis ''B''''n'' of the cyclotomic units can be constructed with the property that ''B''''d'' ⊆ ''B''''n'' for ''d'' | ''n''.〔http://perisic.com/cyclotomic〕 ==See also== *Elliptic unit *Modular unit 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cyclotomic unit」の詳細全文を読む スポンサード リンク
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