Es ist

${\displaystyle {}{\begin{aligned}X^{12}-1&={\left(X^{6}-1\right)}{\left(X^{6}+1\right)}\\&=\Phi _{1}\cdot \Phi _{2}\cdot \Phi _{3}\cdot \Phi _{4}\cdot \Phi _{6}\cdot \Phi _{12}.\end{aligned}}}$

Wegen

${\displaystyle {}X^{6}-1=\Phi _{1}\cdot \Phi _{2}\cdot \Phi _{3}\cdot \Phi _{6}\,}$

muss also

${\displaystyle {}X^{6}+1=\Phi _{4}\cdot \Phi _{12}={\left(X^{2}+1\right)}\cdot \Phi _{12}\,.}$

Polynomdivision ergibt

${\displaystyle {}\Phi _{12}=X^{4}-X^{2}+1\,.}$