Question

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

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

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 Z², such that 2=2x=0.

+	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

Determine multiplication table 

For multiplication, compute the product and then divided by x²+1 and write the remainder

For example,$$\begin{gathered} x.x=x^2=\left(x^2+1\right)-1 \\ thus,x.x=-1=1 \end{gathered}$$

The equality follows since the coefficient is in Z².

g	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.