Cyclotomic unit explained

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 - 1) for ζ 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.^[1]

• If is the power of a prime, then is not a unit; however the numbers for, and ±ζ generate the group of cyclotomic units.
• If is a composite number having two or more distinct prime factors, then is a unit. The subgroup of cyclotomic units generated by with is not of finite index in general.^[2]

The cyclotomic units satisfy distribution relations. Let be a rational number prime to and let denote . Then for we have ^[3]

Using these distribution relations and the symmetry relation a basis B_n of the cyclotomic units can be constructed with the property that for .^[4]

See also

• Elliptic unit
• Modular unit

References

• Book: Lang, Serge . Serge Lang

. Serge Lang . Cyclotomic Fields I and II . second combined . . . 121 . 3-540-96671-4 . 0704.11038 . 1990 .

• Book: Narkiewicz, Władysław . Elementary and analytic theory of numbers . registration . Second, substantially revised and extended . . 3-540-51250-0 . 1990 . 0717.11045 .
• Book: Washington, Lawrence C. . Lawrence C. Washington

. Lawrence C. Washington . Introduction to Cyclotomic Fields . . 0-387-94762-0 . 0966.11047. 2nd . Graduate Texts in Mathematics . 83 . 1997 .

Notes and References

1. Washington, Theorem 8.2
2. Washington, 8.8, page 150, for n equal to 55.
3. Lang (1990) p.157
4. Web site: Marc Conrad's Cyclotomic Units.