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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