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cyclotomic polynomial (Definition)

Definition

For any positive integer $ n$, the $ n$-th cyclotomic polynomial $ \Phi_n(x)$ is defined as

$\displaystyle \Phi_n(x)=\prod_\zeta(x-\zeta), $
where $ \zeta$ ranges over the primitive $ n$-th roots of unity.

Examples

The first few cyclotomic polynomials are as follows:

$\displaystyle \Phi_1(x)$ $\displaystyle =x-1$    
$\displaystyle \Phi_2(x)$ $\displaystyle =x+1$    
$\displaystyle \Phi_3(x)$ $\displaystyle =x^2+x+1$    
$\displaystyle \Phi_4(x)$ $\displaystyle =x^2+1$    
$\displaystyle \Phi_5(x)$ $\displaystyle =x^4+x^3+x^2+x+1$    
$\displaystyle \Phi_6(x)$ $\displaystyle =x^2-x+1$    
$\displaystyle \Phi_7(x)$ $\displaystyle =x^6+x^5+x^4+x^3+x^2+x+1$    
$\displaystyle \Phi_8(x)$ $\displaystyle =x^4+1$    
$\displaystyle \Phi_9(x)$ $\displaystyle =x^6+x^3+1$    
$\displaystyle \Phi_{10}(x)$ $\displaystyle =x^4-x^3+x^2-x+1$    
$\displaystyle \Phi_{11}(x)$ $\displaystyle =x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$    
$\displaystyle \Phi_{12}(x)$ $\displaystyle =x^4-x^2+1$    

The preceding examples may give the impression that the coefficients are always $ -1$, 0 or $ 1$, but this is not true in general. For example,

$\displaystyle \Phi_{105}(x)=\,$ $\displaystyle x^{48}+x^{47}+x^{46}-x^{43}-x^{42}-2x^{41}-x^{40}-x^{39}+x^{36}+x^{35}+x^{34}$    
  $\displaystyle \phantom{x^{48}}+x^{33}+x^{32}+x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}+x^{17}+x^{16}+x^{15}$    
  $\displaystyle \phantom{x^{48}}+x^{14}+x^{13}+x^{12}-x^9-x^8-2x^7-x^6-x^5+x^2+x+1$    

Properties

For every positive integer $ n$, $ \Phi_n(x)$ is an irreducible polynomial of degree $ \phi(n)$ in $ \mathbb{Q}[x]$, and is the minimal polynomial of each primitive $ n$-th root of unity. Here $ \phi(n)$ is Euler's phi function.



"cyclotomic polynomial" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: all one polynomial, factoring all-one polynomials using the grouping method, cyclotomic field, root of unity


Attachments:
proof that the cyclotomic polynomial is irreducible (Proof) by djao
factors of $n$ and $x^n-1$ (Theorem) by pahio
examples of cyclotomic polynomials (Example) by alozano
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Cross-references: Euler's phi function, minimal polynomial, degree, irreducible polynomial, coefficients, integer, positive
There are 5 references to this entry.

This is version 11 of cyclotomic polynomial, born on 2002-04-19, modified 2007-12-29.
Object id is 2852, canonical name is CyclotomicPolynomial.
Accessed 5721 times total.

Classification:
AMS MSC: 11C08 (Number theory :: Polynomials and matrices :: Polynomials)
 11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions)
 11R60 (Number theory :: Algebraic number theory: global fields :: Cyclotomic function fields )
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