Difference between revisions of "Cyclotomic polynomial"
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==Definition== | ==Definition== | ||
− | The cyclotomic polynomials are recursively defined as <math>x^n-1=\ | + | The cyclotomic [[polynomials]] are recursively defined as <math>x^n-1=\prod_{d \vert n} \Phi_n (x)</math>, for <math>n \geq 1</math>. All cyclotomic polynomials are [[irreducible polynomial|irreducible]]. |
+ | |||
+ | ==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>. | ||
+ | |||
+ | 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== | ||
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\Phi_7(x)&=x^6+x^5+\cdots + 1 \\ | \Phi_7(x)&=x^6+x^5+\cdots + 1 \\ | ||
\Phi_8(x)&=x^4+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> | \end{align*}</cmath> | ||
+ | |||
+ | {{stub}} |
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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