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== | ||
Revision as of 16:00, 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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