Question

Describe the congruence class in $\mathbf{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$modulo the polynomial x.

Short answer

There is precisely one conjugacy class of polynomials in$\mathrm{F}\left[\mathrm{x}\right]$ for every element$\mathrm{c}\in \mathrm{F}$ considered as a constant polynomial.

Step-by-step solution

Use exercise 6

This is special case of exercise 6 where$\mathrm{a}=0$.

Use Corollary 5.5

By corollary 5.5 which is says that Let $\mathbf{F}$be a field and$\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$a polynomial of degree n in $\mathit{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$, and consider congruence modulo $\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

If $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\in }\mathit{F}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$and role="math" localid="1654235644989" $\mathit{r}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$is the reminder when $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ is divide by $\mathit{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$, then $\mathbf{\left[}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right]}\mathbf{=}\mathbf{\left[}\mathbf{r}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right]}$.

By above corollary, there is precisely one conjugacy class of polynomials in$F\left[x\right]$ for every element $c\in F$considered as a constant polynomial.