Cyklotomický polynom

Cyklotomický polynom je pojem z oblasti matematiky, přesněji z algebry. Je definován pro všechna nenulová přirozená čísla $n$ jako jednoznačně určený polynom s celočíselnými koeficienty, který je dělitelem polynomu $x^{n}-1$ a není dělitelem $x^{k}-1$ pro žádné $k<n$.

Příklady

$~\Phi _{1}(x)=x-1$

$~\Phi _{2}(x)=x+1$

$~\Phi _{3}(x)=x^{2}+x+1$

$~\Phi _{4}(x)=x^{2}+1$

$~\Phi _{5}(x)=x^{4}+x^{3}+x^{2}+x+1$

$~\Phi _{6}(x)=x^{2}-x+1$

$~\Phi _{7}(x)=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$

$~\Phi _{8}(x)=x^{4}+1$

$~\Phi _{9}(x)=x^{6}+x^{3}+1$

$~\Phi _{10}(x)=x^{4}-x^{3}+x^{2}-x+1$

$~\Phi _{11}(x)=x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$

$~\Phi _{12}(x)=x^{4}-x^{2}+1$

$~\Phi _{13}(x)=x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$

$~\Phi _{14}(x)=x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$

$~\Phi _{15}(x)=x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1$

$~\Phi _{16}(x)=x^{8}+1$

$~\Phi _{17}(x)=x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$

$~\Phi _{18}(x)=x^{6}-x^{3}+1$

$~\Phi _{19}(x)=x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$

$~\Phi _{20}(x)=x^{8}-x^{6}+x^{4}-x^{2}+1$

$~\Phi _{21}(x)=x^{12}-x^{11}+x^{9}-x^{8}+x^{6}-x^{4}+x^{3}-x+1$

$~\Phi _{22}(x)=x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$

${\begin{aligned}\Phi _{23}(x)=&\;x^{22}+x^{21}+x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}\\&{}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\end{aligned}}$

$~\Phi _{24}(x)=x^{8}-x^{4}+1$

$~\Phi _{25}(x)=x^{20}+x^{15}+x^{10}+x^{5}+1$

$~\Phi _{26}(x)=x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$

$~\Phi _{27}(x)=x^{18}+x^{9}+1$

$~\Phi _{28}(x)=x^{12}-x^{10}+x^{8}-x^{6}+x^{4}-x^{2}+1$

${\begin{aligned}\Phi _{29}(x)=&\;x^{28}+x^{27}+x^{26}+x^{25}+x^{24}+x^{23}+x^{22}+x^{21}+x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}\\&{}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\end{aligned}}$

$~\Phi _{30}(x)=x^{8}+x^{7}-x^{5}-x^{4}-x^{3}+x+1$

Reference

V tomto článku byl použit překlad textu z článku Cyclotomic polynomial na anglické Wikipedii.