Cyclotomic unit

Short description: Algebraic number field 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 (ζan − 1) for ζ_n an n^th 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 ζan − 1 is not a unit; however the numbers (ζan − 1)/(ζ_n − 1) for (a, n) = 1, and ±ζan generate the group of cyclotomic units.

• If n is a composite number having two or more distinct prime factors, then ζan − 1 is a unit. The subgroup of cyclotomic units generated by (ζan − 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 [math]\displaystyle{ \prod_{p b=a} g_b = g_a }[/math].

Using these distribution relations and the symmetry relation ζan − 1 = −ζan (ζ−an − 1) a basis B_n of the cyclotomic units can be constructed with the property that B_d ⊆ B_n for d | n.

See also

• Elliptic unit
• Modular unit

References

• Lang, Serge (1990). Cyclotomic Fields I and II. Graduate Texts in Mathematics. 121 (second combined ed.). Springer Verlag. ISBN 3-540-96671-4. 
• Narkiewicz, Władysław (1990). Elementary and analytic theory of numbers (Second, substantially revised and extended ed.). Springer-Verlag. ISBN 3-540-51250-0. https://archive.org/details/elementaryanalyt0000nark. 
• Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. 83 (2nd ed.). Springer-Verlag. ISBN 0-387-94762-0.