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# Endlicher Körper/9/Operationstafeln

Wir nehmen die Darstellung

${\displaystyle {\mathbb {F} }_{9}\cong {\mathbb {F} }_{3}[X]/(X^{2}+1)}$.

Dabei ist ${\displaystyle X^{2}+1}$ irreduzibel in ${\displaystyle {\mathbb {F} }_{3}[X]}$, da es keine Nullstelle in ${\displaystyle {\mathbb {F} }_{3}}$ besitzt (da ${\displaystyle -1}$ kein Quadrat in ${\displaystyle {\mathbb {F} }_{3}[X]}$ ist). Es bezeichne ${\displaystyle x}$ die Restklasse von ${\displaystyle X}$.

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

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