Cyclotomic unit


In mathematics, a cyclotomic unit is a unit of an algebraic number field which is the product of numbers of the form for ζ an nth root of unity and 0 < a < n.

Properties

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 within the full real unit group is equal to the class number of the maximal real subfield of the cyclotomic field.
If n is the power of a prime, then ζ − 1 is not a unit; however the numbers / for = 1, and ±ζ generate the group of cyclotomic units in this case.
If n is a composite number, the subgroup of cyclotomic units generated by /with = 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−1. Then for a≠ 0 we have.
Using these distribution relations and the symmetry relation ζ − 1 = -ζ a basis Bn of the cyclotomic units can be constructed with the property that
Bd ⊆ Bn for d | n.