Editing Cyclotomic polynomial (section)

==Examples==

If ''n'' is a [[prime number]], then 
:<math>\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{k=0}^{n-1} x^k.</math>
If ''n'' = 2''p'' where ''p'' is a [[prime number]] other than 2, then
:<math>\Phi_{2p}(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{k=0}^{p-1} (-x)^k.</math>

For ''n'' up to 30, the cyclotomic polynomials are:<ref>{{Cite OEIS|A013595|mode=cs2}}</ref>

:<math>\begin{align}
\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 \\
\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} \\
& \qquad\quad + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \\
\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 \\
\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} \\
& \qquad\quad + 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_{30}(x) &= x^8 + x^7 - x^5 - x^4 - x^3 + x + 1.
\end{align}</math>

The case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a [[coefficient]] other than 1, 0, or −1:<ref>{{citation
 | last = Brookfield | first = Gary
 | doi = 10.4169/math.mag.89.3.179
 | issue = 3
 | journal = Mathematics Magazine
 | jstor = 10.4169/math.mag.89.3.179
 | mr = 3519075
 | pages = 179–188
 | title = The coefficients of cyclotomic polynomials
 | volume = 89
 | year = 2016}}</ref>

:<math>\begin{align}
\Phi_{105}(x) ={}&x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} + x^{36} + x^{35} + x^{34} \\ 
&{}+ x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\
&{}+ x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1. 
\end{align}</math>