cyclotomic polynomial

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cyclotomic polynomial

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Noun

cyclotomic polynomial (plural cyclotomic polynomials)

1. (algebra) For a positive integer n, a polynomial whose roots are the primitive n^th roots of unity, so that its degree is Euler's totient function of n. That is, letting ${\displaystyle \zeta _{n}=e^{i2\pi /n}}$ be the first primitive n^th root of unity, then ${\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq m<n}{\gcd(n,m)=1}}(x-\zeta _{n}^{m})}$ is the n^th such polynomial.

   For a prime number ${\displaystyle p}$, the ${\displaystyle p}$^th cyclotomic polynomial is ${\displaystyle {x^{p}-1 \over x-1}=x^{p-1}+x^{p-2}+...+x^{2}+x+1}$.

   Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing ${\displaystyle x}$ with ${\displaystyle x+1}$.