Definition:Cyclotomic Polynomial
Contents
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 $n$.
Consider whether this needs to be extracted into a second definition, with a formal equivalence proof using Condition for Complex Root of Unity to be Primitive as its basis.
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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:
- $\Phi_1 \left({x}\right) = x - 1$
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.
