Difference between revisions of "Cyclotomic polynomial"

Revision as of 14:18, 31 May 2020

Definition

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

Roots

The roots of $\Phi_n(x)$ are $e^{\frac{2\pi i d}{n}}$ , where $\gcd(d, n) = 1$ . For this reason, due to the Fundamental Theorem of Algebra, we have $\Phi_n(x) = \prod_{d: \gcd(d, n) = 1} (x - e^{\frac{2\pi i d}{n}})$ .

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: $\begin{align*} \Phi_1(x)&=x-1 \\ \Phi_2(x)&=x+1 \\ \Phi_3(x)&=x^2+x+1 \\ \Phi_4(x)&=x^2+1 \\ \Phi_5(x)&=x^4+x^3+x^2+x+1 \\ \Phi_6(x)&=x^2-x+1 \\ \Phi_7(x)&=x^6+x^5+\cdots + 1 \\ \Phi_8(x)&=x^4+1 \\ \Phi_9(x)&=x^6+x^3+1 \\ \Phi_10(x)&=x^4-x^3+x^2-x+1\\ \end{align*}$

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Difference between revisions of "Cyclotomic polynomial"

Revision as of 14:18, 31 May 2020

Definition

Roots

Examples

@@ Line 18: / Line 18: @@
 \Phi_7(x)&=x^6+x^5+\cdots + 1 \\
 \Phi_8(x)&=x^4+1 \\
+\Phi_9(x)&=x^6+x^3+1 \\
+\Phi_10(x)&=x^4-x^3+x^2-x+1\\
 \end{align*}</cmath>
 {{stub}}