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Cyclotomic polynomial - Wikipedia

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In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of 𝑥 𝑛 − 1 and is not a divisor of 𝑥 𝑘 − 1 for any k < n. Its roots are all nth primitive roots of unity 𝑒 2 𝑖 𝜋 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 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 ( 𝑒 2 𝑖 𝜋 / 𝑛 is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is showing that x is a root of 𝑥 𝑛 − 1 if and only if it is a d th primitive root of unity for some d that divides n.[1] If n is a prime number, then If n = 2p where p is a prime number other than 2, then For n up to 30, the cycloto

Cyclotomic polynomial - Wikipedia Jump to content From Wikipedia, the free encyclopedia Irreducible polynomial whose roots are nth roots of unity In mathematics , the n {\displaystyle n} -th cyclotomic polynomial , for any positive integer n {\displaystyle n} , is the unique irreducible polynomial with integer coefficients that is a divisor of x n − 1 {\displaystyle x^{n}-1} and is not a divisor of x k − 1 {\displaystyle x^{k}-1} for any k < n {\displaystyle k<n} . Its roots are all n {\displaystyle n} -th primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k

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