Endliche Körper/Nicht Primkörper/Einige Operationstafeln

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Operationstafeln für ${\displaystyle {\mathbb {F} }_{4}}$

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 ${\displaystyle X}$.

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

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

Operationstafeln für ${\displaystyle {\mathbb {F} }_{8}}$

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 ${\displaystyle X}$.

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

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

Operationstafeln für ${\displaystyle {\mathbb {F} }_{9}}$

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 ${\displaystyle X}$.

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

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

Operationstafeln für ${\displaystyle {\mathbb {F} }_{16}}$

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 ${\displaystyle X}$.

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

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

Operationstafeln für ${\displaystyle {\mathbb {F} }_{25}}$

Wir nehmen die Darstellung

${\displaystyle {\mathbb {F} }_{25}\cong {\mathbb {F} }_{5}[X]/(X^{2}+X+2)}$.

Dabei ist ${\displaystyle X^{2}+X+2}$ irreduzibel in ${\displaystyle {\mathbb {F} }_{5}[X]}$, da es keine Nullstelle in ${\displaystyle {\mathbb {F} }_{5}}$ besitzt. Es bezeichne ${\displaystyle x}$ die Restklasse von ${\displaystyle X}$.

 ${\displaystyle +}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x+1}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$