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ID: INPUT{id.yaml} Title: > Cyclotomic polynomials Definition: > The $n$'th cyclotomic polynomial $\Phi_n(x)$ is the monic polynomial whose roots are the primitive $n$'th roots of unity. Parameters: n: type: Z constraints: $n \geq 1$ Comments: Formulas: formula-via-roots: > $\Phi_n(x) = \prod_{1\leq k \leq n: \gcd(n,k)=1} (x-\exp(2\pi ik/n))$. Programs: program-sage: language: Sage code: | polynomials = {n: cyclotomic_polynomial(n) for n in [1..100]} References: Links: Wiki: title: "Wikipedia: Cyclotomic polynomial" url: https://en.wikipedia.org/wiki/Cyclotomic_polynomial Similar tables: Keywords: Tags: - Polynomial Data properties: type: Z[] complete: no Display properties: number-header: $\Phi_n(x)$ Data: INPUT{polynomials.yaml}
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Cyclotomic polynomials
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Polynomials
$n$
$\Phi_n(x)$
INPUT{polynomials.yaml} (not shown in preview)
Definition
The $n$'th cyclotomic polynomial $\Phi_n(x)$ is the monic polynomial whose roots are the primitive $n$'th roots of unity.
Parameters
$n$
— integer ($n \geq 1$)
Formulas
(1)
$\Phi_n(x) = \prod_{1\leq k \leq n: \gcd(n,k)=1} (x-\exp(2\pi ik/n))$.
Programs
(P1)
Sage
polynomials = {n: cyclotomic_polynomial(n) for n in [1..100]}
Links
[1]
Wikipedia: Cyclotomic polynomial
Data properties
Entries are of type: integral polynomial
Table is complete: no
