# Endlicher Körper/8/Operationstafeln

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Wir nehmen die Darstellung

${\displaystyle {\mathbb {F} }_{8}\cong {\mathbb {F} }_{2}[X]/(X^{3}+X+1)}$.

Dabei ist ${\displaystyle X^{3}+X+1}$ irreduzibel in ${\displaystyle {\mathbb {F} }_{2}[X]}$, da es keine Nullstelle in ${\displaystyle {\mathbb {F} }_{2}}$ besitzt (${\displaystyle 0^{3}+0+1=1}$ und ${\displaystyle 1^{3}+1+1=1}$, also beide ${\displaystyle \neq 0}$). Es bezeichne ${\displaystyle x}$ die Restklasse von ${\displaystyle X}$.

 ${\displaystyle +}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1+x}$ ${\displaystyle x}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle 1+x}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1+x}$ ${\displaystyle x}$ ${\displaystyle x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 0}$

 ${\displaystyle \cdot }$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle 1+x}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x}$ ${\displaystyle 0}$ ${\displaystyle x}$ ${\displaystyle x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle 1}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle 0}$ ${\displaystyle 1+x}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x^{2}}$ ${\displaystyle 0}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle x}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1}$ ${\displaystyle x^{2}}$ ${\displaystyle x}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle x+x^{2}}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle x+x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 1}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle 1+x}$ ${\displaystyle x}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle 1+x+x^{2}}$ ${\displaystyle 1+x^{2}}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle x+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle 1+x}$