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?

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

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 bracket.

In addition, the coefficient is in${\mathit{\mathbb{Z}}}_{\mathbf{3}}$ such that$\mathbf{3}\mathbf{=}\mathbf{3}\mathit{x}\mathbf{=}\mathbf{0}$

Determine multiplication table

For multiplication, compute the product and then divided by ${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

and write the remainder. For example, $\mathbf{2}\mathit{x}\mathbf{\cdot }\mathbf{2}\mathit{x}\mathbf{=}\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{=}\mathbf{4}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{-}\mathbf{4}$; thus,$\mathbf{2}\mathit{x}\mathbf{\cdot }\mathbf{2}\mathit{x}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{=}\mathbf{2}$

The equality follows since the coefficient is in${\mathit{\mathbb{Z}}}_{\mathbf{3}}$

Therefore, the addition and multiplication are constructed by direct computation.