Difference between revisions of "Cyclotomic polynomial"
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\Phi_8(x)&=x^4+1 \\ | \Phi_8(x)&=x^4+1 \\ | ||
\Phi_9(x)&=x^6+x^3+1 \\ | \Phi_9(x)&=x^6+x^3+1 \\ | ||
− | \Phi_ | + | \Phi_{10}(x)&=x^4-x^3+x^2-x+1\\ |
\end{align*}</cmath> | \end{align*}</cmath> | ||
{{stub}} | {{stub}} |
Revision as of 16:00, 31 May 2020
Definition
The cyclotomic polynomials are recursively defined as , for
. All cyclotomic polynomials are irreducible.
Roots
The roots of are
, where
. For this reason, due to the Fundamental Theorem of Algebra, we have
.
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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