Endliche Körper/Nicht Primkörper/Einige Operationstafeln

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

Wir nehmen die Darstellung

{\mathbb {F} }_{4}\cong {\mathbb {F} }_{2}[X]/(X^{2}+X+1)

.

Dabei ist $X^{2}+X+1$ irreduzibel in ${\mathbb {F} }_{2}[X]$ , da es vom Grad 2 ist und keine Nullstelle in ${\mathbb {F} }_{2}$ besitzt. Es bezeichne $x$ die Restklasse von $X$ .

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

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

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

Wir nehmen die Darstellung

{\mathbb {F} }_{8}\cong {\mathbb {F} }_{2}[X]/(X^{3}+X+1)

.

Dabei ist $X^{3}+X+1$ irreduzibel in ${\mathbb {F} }_{2}[X]$ , da es keine Nullstelle in ${\mathbb {F} }_{2}$ besitzt ( $0^{3}+0+1=1$ und $1^{3}+1+1=1$ , also beide $\neq 0$ ). Es bezeichne $x$ die Restklasse von $X$ .

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

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

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 $X$ .

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

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

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

Wir nehmen die Darstellung

{\mathbb {F} }_{16}\cong {\mathbb {F} }_{2}[X]/(X^{4}+X+1)

.

Dabei ist $X^{4}+X+1$ irreduzibel in ${\mathbb {F} }_{2}[X]$ , da es keine Nullstelle in ${\mathbb {F} }_{2}$ besitzt. Es bezeichne $x$ die Restklasse von $X$ .

+

0

1

x

x+1

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

0

0

1

x

x+1

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

1

1

0

x+1

x

x^{2}+1

x^{2}

x^{2}+x+1

x^{2}+x

x^{3}+1

x^{3}

x^{3}+x+1

x^{3}+x

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x

x

x+1

0

1

x^{2}+x

x^{2}+x+1

x^{2}

x^{2}+1

x^{3}+x

x^{3}+x+1

x^{3}

x^{3}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x+1

x+1

x

1

0

x^{2}+x+1

x^{2}+x

x^{2}+1

x^{2}

x^{3}+x+1

x^{3}+x

x^{3}+1

x^{3}

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{2}

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

0

1

x

x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{2}+1

x^{2}+1

x^{2}

x^{2}+x+1

x^{2}+x

1

0

x+1

x

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+1

x^{3}

x^{3}+x+1

x^{3}+x

x^{2}+x

x^{2}+x

x^{2}+x+1

x^{2}

x^{2}+1

x

x+1

0

1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x

x^{3}+x+1

x^{3}

x^{3}+1

x^{2}+x+1

x^{2}+x+1

x^{2}+x

x^{2}+1

x^{2}

x+1

x

1

0

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{3}+x+1

x^{3}+x

x^{3}+1

x^{3}

x^{3}

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

0

1

x

x+1

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

x^{3}+1

x^{3}+1

x^{3}

x^{3}+x+1

x^{3}+x

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

1

0

x+1

x

x^{2}+1

x^{2}

x^{2}+x+1

x^{2}+x

x^{3}+x

x^{3}+x

x^{3}+x+1

x^{3}

x^{3}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x

x+1

0

1

x^{2}+x

x^{2}+x+1

x^{2}

x^{2}+1

x^{3}+x+1

x^{3}+x+1

x^{3}+x

x^{3}+1

x^{3}

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+x^{2}+1

x^{3}+x^{2}

x+1

x

1

0

x^{2}+x+1

x^{2}+x

x^{2}+1

x^{2}

x^{3}+x^{2}

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

0

1

x

x+1

x^{3}+x^{2}+1

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+1

x^{3}

x^{3}+x+1

x^{3}+x

x^{2}+1

x^{2}

x^{2}+x+1

x^{2}+x

1

0

x+1

x

x^{3}+x^{2}+x

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x

x^{3}+x+1

x^{3}

x^{3}+1

x^{2}+x

x^{2}+x+1

x^{2}

x^{2}+1

x

x+1

0

1

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{3}+x+1

x^{3}+x

x^{3}+1

x^{3}

x^{2}+x+1

x^{2}+x

x^{2}+1

x^{2}

x+1

x

1

0

\cdot

0

1

x

x+1

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

x

x+1

x^{2}

x^{2}+1

x^{2}+x

x^{2}+x+1

x^{3}

x^{3}+1

x^{3}+x

x^{3}+x+1

x^{3}+x^{2}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

x

0

x

x^{2}

x^{2}+x

x^{3}

x^{3}+x

x^{3}+x^{2}

x^{3}+x^{2}+x

x+1

1

x^{2}+x+1

x^{2}+1

x^{3}+x+1

x^{3}+1

x^{3}+x^{2}+x+1

x^{3}+x^{2}+1

x+1

0

x+1

x^{2}+x

x^{2}+1

x^{3}+x^{2}

x^{3}+x^{2}+x+1

x^{3}+x

x^{3}+1

x^{3}+x+1

x^{3}

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{2}+x+1

x^{2}

1

x

x^{2}

0

x^{2}

x^{3}

x^{3}+x^{2}

x+1

x^{2}+x+1

x^{3}+x+1

x^{3}+x^{2}+x+1

x^{2}+x

x

x^{3}+x^{2}+x

x^{3}+x

x^{2}+1

1

x^{3}+x^{2}+1

x^{3}+1

x^{2}+1

0

x^{2}+1

x^{3}+x

x^{3}+x^{2}+x+1

x^{2}+x+1

x

x^{3}+x^{2}+1

x^{3}

x^{3}+x^{2}+x

x^{3}+x+1

x^{2}

1

x^{3}+1

x^{3}+x^{2}

x+1

x^{2}+x

x^{2}+x

0

x^{2}+x

x^{3}+x^{2}

x^{3}+x

x^{3}+x+1

x^{3}+x^{2}+1

x^{2}+x+1

1

x^{2}+1

x+1

x^{3}+1

x^{3}+x^{2}+x+1

x^{3}+x^{2}+x

x^{3}

x

x^{2}

x^{2}+x+1

0

x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+1

x^{3}+x^{2}+x+1

x^{3}

1

x^{2}+x

x^{3}+x^{2}+1

x^{3}+x

x+1

x^{2}

x

x^{2}+1

x^{3}+x^{2}

x^{3}+x+1

x^{3}

0

x^{3}

x+1

x^{3}+x+1

x^{2}+x

x^{3}+x^{2}+x

x^{2}+1

x^{3}+x^{2}+1

x^{3}+x^{2}

x^{2}

x^{3}+x^{2}+x+1

x^{2}+x+1

x^{3}+x

x

x^{3}+1

1

x^{3}+1

0

x^{3}+1

1

x^{3}

x

x^{3}+x+1

x+1

x^{3}+x

x^{2}

x^{3}+x^{2}+1

x^{2}+1

x^{3}+x^{2}

x^{2}+x

x^{3}+x^{2}+x+1

x^{2}+x+1

x^{3}+x^{2}+x

x^{3}+x

0

x^{3}+x

x^{2}+x+1

x^{3}+x^{2}+1

x^{3}+x^{2}+x

x^{2}

x^{3}+1

x+1

x^{3}+x^{2}+x+1

x^{2}+1

x^{3}

x

1

x^{3}+x+1

x^{2}+x

x^{3}+x^{2}

x^{3}+x+1

0

x^{3}+x+1

x^{2}+1

x^{3}+x^{2}+x

x^{3}+x

1

x^{3}+x^{2}+x+1

x^{2}

x^{2}+x+1

x^{3}+x^{2}

x

x^{3}+1

x^{3}+x^{2}+1

x^{2}+x

x^{3}

x+1

x^{3}+x^{2}

0

x^{3}+x^{2}

x^{3}+x+1

x^{2}+x+1

x^{2}+1

x^{3}+1

x^{3}+x^{2}+x

x

x^{3}+x

x^{2}+x

1

x^{3}+x^{2}+1

x^{3}+x^{2}+x+1

x+1

x^{2}

x^{3}

x^{3}+x^{2}+1

0

x^{3}+x^{2}+1

x^{3}+1

x^{2}

1

x^{3}+x^{2}

x^{3}

x^{2}+1

x

x^{3}+x^{2}+x+1

x^{3}+x+1

x^{2}+x

x+1

x^{3}+x^{2}+x

x^{3}+x

x^{2}+x+1

x^{3}+x^{2}+x

0

x^{3}+x^{2}+x

x^{3}+x^{2}+x+1

1

x^{3}+x^{2}+1

x+1

x

x^{3}+x^{2}

x^{3}+1

x^{2}+x+1

x^{2}+x

x^{3}

x^{2}

x^{3}+x

x^{3}+x+1

x^{2}+1

x^{3}+x^{2}+x+1

0

x^{3}+x^{2}+x+1

x^{3}+x^{2}+1

x

x^{3}+1

x^{2}+x

x^{2}

x^{3}+x+1

1

x^{3}+x^{2}+x

x^{3}+x^{2}

x+1

x^{3}

x^{2}+x+1

x^{2}+1

x^{3}+x

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

Wir nehmen die Darstellung

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

.

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

+

0

1

2

3

4

x

x+1

x+2

x+3

x+4

2x

2x+1

2x+2

2x+3

2x+4

3x

3x+1

3x+2

3x+3

3x+4

4x

4x+1

4x+2

4x+3

4x+4

0

0

1

2

3

4

x

x+1

x+2

x+3

x+4

2x

2x+1

2x+2

2x+3

2x+4

3x

3x+1

3x+2

3x+3

3x+4

4x

4x+1

4x+2

4x+3

4x+4

1

1

2

3

4

0

x+1

x+2

x+3

x+4

x

2x+1

2x+2

2x+3

2x+4

2x

3x+1

3x+2

3x+3

3x+4

3x

4x+1

4x+2

4x+3

4x+4

4x

2

2

3

4

0

1

x+2

x+3

x+4

x

x+1

2x+2

2x+3

2x+4

2x

2x+1

3x+2

3x+3

3x+4

3x

3x+1

4x+2

4x+3

4x+4

4x

4x+1

3

3

4

0

1

2

x+3

x+4

x

x+1

x+2

2x+3

2x+4

2x

2x+1

2x+2

3x+3

3x+4

3x

3x+1

3x+2

4x+3

4x+4

4x

4x+1

4x+2

4

4

0

1

2

3

x+4

x

x+1

x+2

x+3

2x+4

2x

2x+1

2x+2

2x+3

3x+4

3x

3x+1

3x+2

3x+3

4x+4

4x

4x+1

4x+2

4x+3

x

x

x+1

x+2

x+3

x+4

2x

2x+1

2x+2

2x+3

2x+4

3x

3x+1

3x+2

3x+3

3x+4

4x

4x+1

4x+2

4x+3

4x+4

0

1

2

3

4

x+1

x+1

x+2

x+3

x+4

x

2x+1

2x+2

2x+3

2x+4

2x

3x+1

3x+2

3x+3

3x+4

3x

4x+1

4x+2

4x+3

4x+4

4x

1

2

3

4

0

x+2

x+2

x+3

x+4

x

x+1

2x+2

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