# Difference between revisions of "Cyclotomic polynomial"

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==Roots== | ==Roots== | ||

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

+ | |||

+ | Therefore, <math>x^{n}-1</math> can be factored as <math>\Phi_{d_{1}}x*\Phi_{d_{2}}x*\Phi_{d_{3}}x*\cdots*\Phi_{d_{k}}</math> where <math>d_1, d_2, d_3\cdots, d_k</math> are the positive divisors of <math>n</math>. | ||

==Examples== | ==Examples== |

## Latest revision as of 17:00, 31 May 2020

## Definition

The cyclotomic polynomials are recursively defined as , for . All cyclotomic polynomials are irreducible.

## Roots

The roots of are , where . For this reason, due to the Fundamental Theorem of Algebra, we have .

Therefore, can be factored as where are the positive divisors of .

## Examples

For a prime , , because for a prime , and so we can factorise to obtain the required result.

The first few cyclotomic polynomials are as shown:

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