Question

In Exercises 1-4, write out the addition and multiplication tables for the congruence class ring F $\mathbf{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{/}\mathbf{\left(}\mathbf{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right)}$. In each case, is$\mathbf{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{/}\mathbf{\left(}\mathbf{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right)}$ a field?$\mathit{F}\mathbf{=}{\mathit{\mathbb{Z}}}_{\mathbf{2}}\mathbf{;}\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

Short answer

The addition and multiplication are constructed by direct computation.

Step-by-step solution

Determine addition table

For simplification, all the classes are written without a bracket.

In addition, the coefficient is in ${\mathbb{Z}}_2$, such that $2=2x=0$.

+	0	1	$x$	$x+1$
0	0	1	role="math" localid="1654237771067" $x$	$x+1$
1	1	0	$x+1$	role="math" localid="1654237777206" $x$
$x$	$x$	$x+1$	0	1
$x+1$	$x+1$	$x$	1	0

Determine multiplication table

For multiplication, compute the product and then divided by $x^2+1$and write the remainder.

For example, $x\cdot x=x^2=(x^2+1)-1$; thus, $x\cdot x=-1=1$.

The equality follows since the coefficient is in${\mathbb{Z}}_2$.

$\mathbf{·}$	0	1	$x$	$x+1$
0	0	0	0	0
1	0	1	$x$	$x+1$
$x$	0	$x$	1	$x+1$
$x+1$	0	$x+1$	$x+1$	0

Thus, addition and multiplication are constructed by direct computation.