cyclotomic field
A cyclotomic field (or cyclotomic number field) is a cyclotomic extension of . These are all of the form , where is a primitive th root of unity (http://planetmath.org/PrimitiveNthRootOfUnity).
The ring of integers of a cyclotomic field always has a power basis over (http://planetmath.org/PowerBasisOverMathbbZ). Specifically, the ring of integers of is .
Given a , its minimal polynomial over is the cyclotomic polynomial . Thus, , where denotes the Euler phi function.
If is odd, then . There are many ways to prove this, but the following is a relatively short proof: Since , we have that . We also have that . Thus, . It follows that .
Note. If is a positive integer and is an integer such that , then and are the same cyclotomic field.
Title | cyclotomic field |
---|---|
Canonical name | CyclotomicField |
Date of creation | 2013-03-22 17:10:25 |
Last modified on | 2013-03-22 17:10:25 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 9 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 11R18 |
Classification | msc 11-00 |
Synonym | cyclotomic number field |
Related topic | CyclotomicExtension |
Related topic | CyclotomicPolynomial |