${\displaystyle {}\zeta }$

${\displaystyle {}\zeta _{8}^{4}=-1\,.}$

Somit wird ${\displaystyle {}X^{4}+1}$ von ${\displaystyle {}\zeta _{8}}$ annulliert, und da der Grad der Körpererweiterung ${\displaystyle {}\mathbb {Q} \subseteq L}$ gleich ${\displaystyle {}4}$ ist, muss ${\displaystyle {}X^{4}+1}$ das Minimalpoynom sein.

${\displaystyle {}\zeta _{8}^{2}={\mathrm {i} }\,}$

und da ${\displaystyle {}\mathbb {Q} [{\mathrm {i} }]\subseteq L}$ den Grad ${\displaystyle {}2}$ besitzt, ist ${\displaystyle {}X^{2}-{\mathrm {i} }}$ das Minimalpolynom von ${\displaystyle {}\zeta _{8}}$ über ${\displaystyle {}\mathbb {Q} [{\mathrm {i} }]}$.

${\displaystyle {}\mathbb {Q} [{\sqrt {2}}]\subseteq L}$

${\displaystyle {}2}$

${\displaystyle {}2}$

${\displaystyle {}\zeta _{8}^{2}={\mathrm {i} }}$

${\displaystyle {}\zeta _{8}^{2}-{\sqrt {2}}\zeta _{8}={\mathrm {i} }-{\sqrt {2}}\cdot {\frac {1}{2}}{\left({\sqrt {2}}+{\sqrt {2}}{\mathrm {i} }\right)}=-1\,}$

und somit ist ${\displaystyle {}X^{2}+{\sqrt {2}}X+1}$ das Minimalpolynom von ${\displaystyle {}\zeta _{8}}$ über ${\displaystyle {}\mathbb {Q} [{\sqrt {2}}]}$.