Benutzer:Abrankov/Vorlesung/Endliche Körper/Nicht Primkörper/Einige Operationstafeln

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

Wir nehmen die Darstellung

${\mathbb {F} }_{9}\cong {\mathbb {F} }_{3}[X]/(X^{2}+1)$.

Dabei ist $X^{2}+1$ irreduzibel in ${\mathbb {F} }_{3}[X]$, da es keine Nullstelle in ${\mathbb {F} }_{3}$ besitzt (da $-1$ kein Quadrat in ${\mathbb {F} }_{3}[X]$ ist). Es bezeichne $x$ die Restklasse von $X$.

$+$	$0$	$1$	$2$	$x$	$x+1$	$x+2$	$2x$	$2x+1$	$2x+2$
$0$	$0$	$1$	$2$	$x$	$x+1$	$x+2$	$2x$	$2x+1$	$2x+2$
$1$	$1$	$2$	$0$	$x+1$	$x+2$	$x$	$2x+1$	$2x+2$	$2x$
$2$	$2$	$0$	$1$	$x+2$	$x$	$x+1$	$2x+2$	$2x$	$2x+1$
$x$	$x$	$x+1$	$x+2$	$2x$	$2x+1$	$2x+2$	$0$	$1$	$2$
$x+1$	$x+1$	$x+2$	$x$	$2x+1$	$2x+2$	$2x$	$1$	$2$	$0$
$x+2$	$x+2$	$x$	$x+1$	$2x+2$	$2x$	$2x+1$	$2$	$0$	$1$
$2x$	$2x$	$2x+1$	$2x+2$	$0$	$1$	$2$	$x$	$x+1$	$x+2$
$2x+1$	$2x+1$	$2x+2$	$2x$	$1$	$2$	$0$	$x+1$	$x+2$	$x$
$2x+2$	$2x+2$	$2x$	$2x+1$	$2$	$0$	$1$	$x+2$	$x$	$x+1$

$\cdot$	$0$	$1$	$2$	$x$	$x+1$	$x+2$	$2x$	$2x+1$	$2x+2$
$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$1$	$0$	$1$	$2$	$x$	$x+1$	$x+2$	$2x$	$2x+1$	$2x+2$
$2$	$0$	$2$	$1$	$2x$	$2x+2$	$2x+1$	$x$	$x+2$	$x+1$
$x$	$0$	$x$	$2x$	$2$	$x+2$	$2x+2$	$1$	$x+1$	$2x+1$
$x+1$	$0$	$x+1$	$2x+2$	$x+2$	$2x$	$1$	$2x+1$	$2$	$x$
$x+2$	$0$	$x+2$	$2x+1$	$2x+2$	$1$	$x$	$x+1$	$2x$	$2$
$2x$	$0$	$2x$	$x$	$1$	$2x+1$	$x+1$	$2$	$2x+2$	$x+2$
$2x+1$	$0$	$2x+1$	$x+2$	$x+1$	$2$	$2x$	$2x+2$	$x$	$1$
$2x+2$	$0$	$2x+2$	$x+1$	$2x+1$	$x$	$2$	$x+2$	$1$	$2x$