Cyclotomic polynomial
Definition
The cyclotomic polynomials are recursively defined as , for . All cyclotomic polynomials are irreducible over the rationals.
Roots
The roots of are , where . For this reason, due to the Fundamental Theorem of Algebra, we have .
Therefore, can be factored as where are the positive divisors of .
Examples
For a prime , , because for a prime , and so we can factorise to obtain the required result.
The first few cyclotomic polynomials are as shown:
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