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 ${\displaystyle X}$.

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

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