Question

In Exercises 1-4, write out the addition and multiplication tables for the congruence class ring $F\left[x\right]/\left(p\left(x\right)\right)$localid="1649059931959">FF[x]/(px). In e ach case, is $\mathit{F}\left[x\right]\mathbf{/}\left(p\left(x\right)\right)$a field?

localid="1649063028915" $\mathit{F}\mathbf{=}{\mathit{Z}}_{\mathbf{2}}\mathbf{;}\mathbf{}\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathit{x}\mathbf{+}\mathbf{1}$

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 bracket.

In addition, the coefficient is in Z², such that $2=2x=2x^2=0$

+	0	1	x	x+1	x2	x2+1	x2+x	x2+x+1
0	0	1	x	x+1	x2	x2+1	x2+x	x2+x+1
1	1	0	x+1	x	x2+1	x2	x2+x+1	x2+x
x	x	x+1	0	1	x2+x	x2+x+1	x2	x2+1
x+1	x+1	x	1	0	x2+x+1	x2+x	x2+1	x2
x2	x2	x2+1	x2+x	x2+x+1	0	1	x	x+1
x2+1	x2+1	x2	x2+x	x2+x+1	1	0	x+1	x
x2+x	x2+x	x2+x+1	x2	x2+1	x	x+1	0	1
x2+x+1	x2+x+1	x2+x	x2+1	x2	x+1	x	1	0

Determine multiplication table

For multiplication, compute the product and then divided by $x^3+x+1$and write the remainder.

For example, $x^2\times x^2=x^4=x\left(x^3+x+1\right)-x^2-x$

thus, $x^2\times x^2=-x^2-x=x^2+x$

The equality follows since the coefficient Z² isin

.	0	1	x	x+1	x2	x2+1	x2+x	x2+x+1
0	0	0	0	0	0	0	0	0
1	0	1	x	x+1	x2	x2	x2+x	x2+x+1
x	0	x	x2	x2+x	x+1	x+1	x2+x+1	x2+1
x+1	0	x+1	x2+x	x+1	x2+x+1	x2+x+1	1	x
x2	0	x2	x+1	x2+x+1	x2+x	x2+x	x2+1	1
x2+1	0	x2+1	1	x2	x2	x2	x+1	x2+x
x2+x	0	x2+x	x2+x+1	1	x2+1	x2+1	x	x2
x2+x+1	0	x2+x+1	x2+1	x	1	1	x2	x+1