Cyclotomic polynomial

Definition

The cyclotomic polynomials are recursively defined as $x^n-1=\prod{d \vert n} \Phi_n (x)$ , for $n \geq 1$ .

Examples

For a prime $p$ , $\Phi_p (x)=x^{p-1}+x^{p-2}+ \cdots + 1$ , because for a prime $p$ , $\Phi_p (x) \cdot \Phi_1 (x)=x^p - 1$ and so we can factorise $x^p - 1$ to obtain the required result.

The first few cyclotomic polynomials are as shown: $\begin{align*} \Phi_1(x)&=x-1 \\ \Phi_2(x)&=x+1 \\ \Phi_3(x)&=x^2+x+1 \\ \Phi_4(x)&=x^2+1 \\ \Phi_5(x)&=x^4+x^3+x^2+x+1 \\ \Phi_6(x)&=x^2-x+1 \\ \Phi_7(x)&=x^6+x^5+\cdots + 1 \\ \Phi_8(x)&=x^4+1 \\ \end{align*}$

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Cyclotomic polynomial

Definition

Examples