Endlicher Körper/16/Operationstafeln

Zur Navigation springen Zur Suche springen

Wir nehmen die Darstellung

${\displaystyle {\mathbb {F} }_{16}\cong {\mathbb {F} }_{2}[X]/(X^{4}+X+1)}$.

Dabei ist ${\displaystyle X^{4}+X+1}$ irreduzibel in ${\displaystyle {\mathbb {F} }_{2}[X]}$, da es keine Nullstelle in ${\displaystyle {\mathbb {F} }_{2}}$ besitzt. Es bezeichne ${\displaystyle x}$ die Restklasse von ${\displaystyle X}$.

 ${\displaystyle +}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x+1}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 0}$

 ${\displaystyle \cdot }$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x}$ ${\displaystyle 0}$ ${\displaystyle x}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x+1}$ ${\displaystyle 1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x+1}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle x^{2}}$ ${\displaystyle 0}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle 0}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}+x}$ ${\displaystyle 0}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle 1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}}$ ${\displaystyle x}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle 1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle 0}$ ${\displaystyle x^{3}}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+1}$ ${\displaystyle 1}$ ${\displaystyle x^{3}}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}+x}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x+1}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{2}}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+x}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle 0}$ ${\displaystyle x^{3}+x^{2}+x+1}$ ${\displaystyle x^{3}+x^{2}+1}$ ${\displaystyle x}$ ${\displaystyle x^{3}+1}$ ${\displaystyle x^{2}+x}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}+x+1}$ ${\displaystyle 1}$ ${\displaystyle x^{3}+x^{2}+x}$ ${\displaystyle x^{3}+x^{2}}$ ${\displaystyle x+1}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{2}+x+1}$ ${\displaystyle x^{2}+1}$ ${\displaystyle x^{3}+x}$