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 ${\displaystyle X}$.

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

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