Cyclotomic unit
Template:Short description 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 (ζTemplate:Su − 1) for ζTemplate:Su 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.[1]
- If Template:Mvar is the power of a prime, then Template:Math is not a unit; however the numbers Template:Math for Template:Math, and ±ζTemplate:Su generate the group of cyclotomic units.
- If Template:Mvar is a composite number having two or more distinct prime factors, then Template:Math is a unit. The subgroup of cyclotomic units generated by Template:Math with Template:Math is not of finite index in general.[2]
The cyclotomic units satisfy distribution relations. Let Template:Mvar be a rational number prime to Template:Mvar and let Template:Math denote Template:Math. Then for Template:Math we have Template:Nowrap[3]
Using these distribution relations and the symmetry relation Template:Math a basis Bn of the cyclotomic units can be constructed with the property that Template:Math for Template:Math.[4]
See also
Notes
References
- ↑ Washington, Theorem 8.2
- ↑ Washington, 8.8, page 150, for n equal to 55.
- ↑ Lang (1990) p.157
- ↑ Template:Cite web