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 (ζ$a n$ &minus; 1) for ζ$n$ 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 (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.


 * If $n$ is the power of a prime, then $ζa n − 1$ is not a unit; however the numbers $(ζa n − 1)/(ζ n − 1)$ for $(a, n) = 1$, and ±ζ$a n$ generate the group of cyclotomic units.


 * If $n$ is a composite number having two or more distinct prime factors, then $ζa n − 1$ is a unit. The subgroup of cyclotomic units generated by $(ζa n − 1)/(ζ n − 1)$ 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 $g_{a}$ denote $exp(2πia) − 1$. Then for $a ≠ 0$ we have $ \prod_{p b=a} g_b = g_a$.

Using these distribution relations and the symmetry relation $ζa n − 1 = −ζa n (ζ−a n − 1)$ a basis Bn of the cyclotomic units can be constructed with the property that $B_{d} ⊆ B_{n}$ for $d | n$.