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number > number theory > algebraic number theory > cyclotomic field > cyclotomic polynomial

mathematical analysis > function > elementary function > polynomial > cyclotomic polynomial

algebra > polynomial > cyclotomic polynomial

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cyclotomic polynomial

In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of ${\\displaystyle x^{n}-1}$ and is not a divisor of ${\\displaystyle x^{k}-1}$ for any k < n. Its roots are all nth primitive roots of unity ${\\displaystyle e^{2i\\pi {\ rac {k}{n}}}}$ , where k runs over the positive integers not greater than n and coprime to n (and i is the imaginary unit). In other words, the nth cyclotomic polynomial is equal to
${\\displaystyle \\Phi _{n}(x)=\\prod _{\\stackrel {1\\leq k\\leq n}{\\gcd(k,n)=1}}\\left(x-e^{2i\\pi {\ rac {k}{n}}}\ ight).}$
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( ${\\displaystyle e^{2i\\pi /n}}$ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is
${\\displaystyle \\prod _{d\\mid n}\\Phi _{d}(x)=x^{n}-1,}$
showing that $x$ is a root of ${\\displaystyle x^{n}-1}$ if and only if it is a d th primitive root of unity for some d that divides n.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Cyclotomic_polynomial)

polynôme cyclotomique

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