Cyklotomický polynom

Šablona:Neověřeno 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

${\mathrm{\Phi }}_1(x)=x-1$

${\mathrm{\Phi }}_2(x)=x+1$

${\mathrm{\Phi }}_3(x)=x^2+x+1$

${\mathrm{\Phi }}_4(x)=x^2+1$

${\mathrm{\Phi }}_5(x)=x^4+x^3+x^2+x+1$

${\mathrm{\Phi }}_6(x)=x^2-x+1$

${\mathrm{\Phi }}_7(x)=x^6+x^5+x^4+x^3+x^2+x+1$

${\mathrm{\Phi }}_8(x)=x^4+1$

${\mathrm{\Phi }}_9(x)=x^6+x^3+1$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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