Difference between revisions of "Cyclotomic polynomial"

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(New entry, just a stub)
 
imported>Richard Pinch
(subpages)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
{{subpages}}
In [[algebra]], a '''cyclotomic polynomial''' is a [[polynomial]] whose roots are a set of primitive [[root of unity|roots of unity]].  The ''n''-th cyclotomic polynomial, denoted by Φ<sub>''n''</sub> has [[integer]] cofficients.
In [[algebra]], a '''cyclotomic polynomial''' is a [[polynomial]] whose roots are a set of primitive [[root of unity|roots of unity]].  The ''n''-th cyclotomic polynomial, denoted by Φ<sub>''n''</sub> has [[integer]] cofficients.


Line 16: Line 17:


where μ is the [[Möbius function]].
where μ is the [[Möbius function]].
==Examples==
:<math>\Phi_1(X) = X-1  ;\,</math>
:<math>\Phi_2(X) = X+1  ;\,</math>
:<math>\Phi_3(X) = X^2+X+1  ;\,</math>
:<math>\Phi_4(X) = X^2+1  ;\,</math>
:<math>\Phi_5(X) = X^4+X^3+X^2+X+1  ;\,</math>
:<math>\Phi_6(X) = X^2-X+1  ;\,</math>
:<math>\Phi_7(X) = X^6+X^5+X^4+X^3+X^2+X+1  ;\,</math>
:<math>\Phi_8(X) = X^4+1  ;\,</math>
:<math>\Phi_9(X) = X^6+X^3+1  ;\,</math>
:<math>\Phi_{10}(X) = X^4-X^3+X^2-X+1. ;\,</math>

Latest revision as of 14:55, 11 December 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In algebra, a cyclotomic polynomial is a polynomial whose roots are a set of primitive roots of unity. The n-th cyclotomic polynomial, denoted by Φn has integer cofficients.

For a positive integer n, let ζ be a primitive n-th root of unity: then

The degree of is given by the Euler totient function .

Since any n-th root of unity is a primitive d-th root of unity for some factor d of n, we have

By the Möbius inversion formula we have

where μ is the Möbius function.

Examples