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 (ζan − 1) for ζn an nth root of unity and 0 < a < n. Note that if n is the power of a prime ζan − 1 itself is not a unit; however the numbers (ζan − 1)/(ζn − 1) for (a, n) = 1, and ±ζan generate the group of cyclotomic units in this case (n power of a prime). Note also that if n is a composite number, the subgroup of cyclotomic units generated by (ζan − 1)/(ζn − 1)with (a, n) = 1 is not of finite index in general. .