Definition:Cyclotomic Polynomial

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Definition

Let $n \ge 1$ be a natural number.


The $n$th cyclotomic polynomial is the polynomial

$\displaystyle \Phi_n \paren x = \prod_\zeta \paren {x - \zeta}$

where the product runs over all primitive complex $n$th roots of unity, that is, those whose order is $n$.



From Condition for Complex Root of Unity to be Primitive it is seen that this can be expressed as:

$\displaystyle \Phi_n \paren x = \prod_{\substack {1 \mathop \le k \mathop \le n \\ \gcd \set {k, n} = 1} } \paren {x - \exp \paren {\dfrac {2 \pi i k} n} }$


Examples

First Cyclotomic Polynomial

The first cyclotomic polynomial is:

$\map {\Phi_1} x = x - 1$


Cyclotomic Polynomial of Prime Index

Let $p$ be a prime number.

The $p$th cyclotomic polynomial is:

$\Phi_p \left({x}\right) = x^{p-1} + x^{p-2} + \cdots + x + 1$


Also see

  • Results about cyclotomic polynomials can be found here.