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 ${\displaystyle X}$.

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

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