cyclotomic field

A cyclotomic fieldMathworldPlanetmath (or cyclotomic number field) is a cyclotomic extension of . These are all of the form (ωn), where ωn is a primitive nth root of unityMathworldPlanetmath (

The ring of integers of a cyclotomic field always has a power basis over ( Specifically, the ring of integers of (ωn) is [ωn].

Given a ωn, its minimal polynomial over is the cyclotomic polynomialMathworldPlanetmath Φn(x). Thus, [(ωn):]=φ(n), where φ denotes the Euler phi function.

If n is odd, then (ω2n)=(ωn). There are many ways to prove this, but the following is a relatively short proof: Since ωn=ω2n2(ω2n), we have that (ωn)(ω2n). We also have that [(ω2n):]=φ(2n)=φ(2)φ(n)=φ(n)=[(ωn):]. Thus, [(ω2n):(ωn)]=1. It follows that (ω2n)=(ωn).

Note.  If n is a positive integer and m is an integer such that gcd(m,n)=1, then  ωn  and  ωnm  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