We introduce cyclotomic polynomials as minimal polynomials of primitive roots of unity over the field of rational numbers. We show that they have integer coefficients, which are the subject of further analysis. Among other things we find their form through explicit formulas which leads us to some integral identities. We consider the properties of the coefficients of the $n$-th cyclotomic polynomial based on the prime factorization of the number $n$. By introducing an alternative definition of cyclotomic polynomials we prove that their coefficients have no upper bound. Using cyclotomic polynomials we prove Wedderburn’s little theorem.
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