Cyclotomic unit

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 (ζa
n − 1) for ζ
n an n^throot of unity and 0 < a < n. Note that if n is the power of a prime ζa
n − 1 itself is not a unit; however the numbers (ζa
n − 1)/(ζ
n − 1) for (a, n) = 1, and ±ζa
n 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 real 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 (ζa
n − 1)/(ζ
n − 1)with (a, n) = 1 is not of finite index in general.

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