Question

In Exercises 1-4, write out the addition and multiplication tables for the congruence class ring

F$\mathit{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{/}\mathbf{\left(}\mathbf{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right)}$. In each case, is$\mathit{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{/}\mathbf{\left(}\mathbf{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right)}$a field?

$\mathit{F}\mathbf{=}{\mathit{\mathbb{Z}}}_{\mathbf{5}}\mathbf{;}\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

Short answer

The addition and multiplication table will not be fit here.

Step-by-step solution

 Step 1: Determine addition table

Consider the given function $\mathit{F}\mathbf{=}{\mathit{\mathbb{Z}}}_{\mathbf{5}}\mathbf{;}\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

The addition and multiplication table can be $\mathbf{25}\mathbf{\times }\mathbf{25}$table.

The addition table can be calculated as follows:

$\mathbf{[}{\mathbf{a}}_{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\mathbf{x}\mathbf{]}\mathbf{+}\mathbf{[}{\mathbf{b}}_{\mathbf{0}}\mathbf{+}{\mathbf{b}}_{\mathbf{1}}\mathbf{x}\mathbf{]}\mathbf{=}\mathbf{[}\mathbf{(}{\mathbf{a}}_{\mathbf{0}}\mathbf{+}{\mathbf{b}}_{\mathbf{0}}\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{5}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{1}}\mathbf{+}{\mathbf{b}}_{\mathbf{1}}\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{5}\mathbf{)}\mathbf{x}\mathbf{]}$

Determine multiplication table

The multiplication table is written as follows:

$\begin{array}{c}\mathbf{[}{\mathbf{a}}_{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\mathbf{x}\mathbf{]}\mathbf{\cdot }\mathbf{[}{\mathbf{b}}_{\mathbf{0}}\mathbf{+}{\mathbf{b}}_{\mathbf{1}}\mathbf{x}\mathbf{]}\mathbf{=}\mathbf{[}{\mathbf{a}}_{\mathbf{0}}{\mathbf{b}}_{\mathbf{0}}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{0}}{\mathbf{b}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{0}}\mathbf{)}\mathbf{x}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{1}}{\mathbf{x}}^{\mathbf{2}}\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{5}\mathbf{]}\\ \mathbf{=}\mathbf{[}{\mathbf{a}}_{\mathbf{0}}{\mathbf{b}}_{\mathbf{0}}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{0}}{\mathbf{b}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{0}}\mathbf{)}\mathbf{x}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{1}}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{5}\mathbf{]}\\ \mathbf{=}\mathbf{[}\mathbf{(}{\mathbf{a}}_{\mathbf{0}}{\mathbf{b}}_{\mathbf{0}}\mathbf{-}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{1}}\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{5}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{0}}{\mathbf{b}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{0}}\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{5}\mathbf{)}\mathbf{x}\mathbf{]}\end{array}$

As the explicit table $\mathbf{25}\mathbf{\times }\mathbf{25}$ will not be fit here.