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| In mathematics, a '''cyclotomic unit''' is a [[unit (ring theory)|unit]] of an [[algebraic number field]] which is the product of numbers of the form (ζ{{su|b=''n''|p=''a''}} − 1) for ζ{{su|b=''n''}} an ''n''<sup>th</sup> [[root of unity]] and 0 < ''a'' < ''n''. Note that if ''n'' is the power of a prime ζ{{su|b=''n''|p=''a''}} − 1 itself is not a unit; however the numbers (ζ{{su|b=''n''|p=''a''}} − 1)/(ζ{{su|b=''n''}} − 1) for (''a'', ''n'') = 1, and ±ζ{{su|b=''n''|p=''a''}} generate the group of cyclotomic units in this case (''n'' power of a prime).
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| The cyclotomic units form a subgroup of finite [[Index of a subgroup|index]] in the [[Dirichlet's unit theorem|group of units]] of a [[cyclotomic field]]. The index of ths subgroup of ''real'' cyclotomic units (those cyclotomic units in the maximal real subfield) within the full unit group is equal to the [[Class number (number theory)|class number]] of the maximal real subfield of the [[cyclotomic field]].<ref>Washington, Theorem 8.2</ref>
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| Note also that if ''n'' is a [[composite number]], the subgroup of cyclotomic units generated by (ζ{{su|b=''n''|p=''a''}} − 1)/(ζ{{su|b=''n''}} − 1)with (''a'', ''n'') = 1 is not of finite index in general.<ref>Washington, 8.8, page 150, for ''n'' equal to 55.</ref>
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| The cyclotomic units satisfy ''distribution relations''. let ''a'' be a rational number prime to ''p'' and let ''g''<sub>''a''</sub> denote exp(2πi''a'')−1. Then for ''a''≠ 0 we have <math> \prod_{p b=a} g_b = g_a</math>.<ref>Lang (1990) p.157</ref>
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| Using these distribution relations and the symmetry relation ζ{{su|b=''n''|p=''a''}} − 1 = -ζ{{su|b=''n''|p=''a''}} (ζ{{su|b=''n''|p=''-a''}} − 1) a basis ''B''<sub>''n''</sub> of the cyclotomic units can be constructed with the property that
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| ''B''<sub>''d''</sub> ⊆ ''B''<sub>''n''</sub> for ''d'' | ''n''.<ref>http://perisic.com/cyclotomic</ref>
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| ==See also==
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| *[[Elliptic unit]]
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| *[[Modular unit]]
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Cyclotomic Fields I and II | edition=second combined | publisher=[[Springer Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=121 | isbn=3-540-96671-4 | zbl=0704.11038 | year=1990 }}
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| * {{cite book | last=Washington | first=Lawrence C. | authorlink=Lawrence C. Washington | title=Introduction to Cyclotomic Fields | publisher=[[Springer-Verlag]] | isbn=0-387-94762-0 | zbl=0966.11047| edition=2nd | series=Graduate Texts in Mathematics | volume=83 | year=1997 }}
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| [[Category:Algebraic number theory]]
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| [[Category:Cyclotomic fields]]
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| {{numtheory-stub}}
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Latest revision as of 11:10, 7 March 2014
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