# Endlicher Körper/4/Operationstafeln

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Wir nehmen die Darstellung

${\displaystyle {\mathbb {F} }_{4}\cong {\mathbb {F} }_{2}[X]/(X^{2}+X+1)}$.

Dabei ist ${\displaystyle X^{2}+X+1}$ irreduzibel in ${\displaystyle {\mathbb {F} }_{2}[X]}$, da es vom Grad 2 ist und keine Nullstelle in ${\displaystyle {\mathbb {F} }_{2}}$ besitzt. Es bezeichne ${\displaystyle x}$ die Restklasse von ${\displaystyle X}$.

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

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