Question

List all monic irreducible polynomials of degree 2 in ${\mathbf{\mathbb{Z}}}_{\mathbf{3}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$. Do the same in ${\mathbf{\mathbb{Z}}}_{\mathbf{5}}\left[x\right]$.

Short answer

It is proved that the required polynomials can be seen in the answer.

Step-by-step solution

Factor Theorem:

If F is any field such that $\mathit{f}\left(x\right)\mathbf{\in }\mathit{F}\left[x\right]$and $\mathit{k}\mathbf{\in }\mathit{F}$. Then, k will be the root of the polynomial f(x), if and only if$\left(x-k\right)$ is a factor of f(x) in F[X].

Monic Irreducible Polynomial:

As we know, the monic irreducible polynomial of degree 2 is given in the form: $x^2+ax+b ..........\left(b\ne 0\right)$

Now, for ${\mathrm{\mathbb{Z}}}_3\left[x\right]$, the monic irreducible polynomials are:

$\left\{\left(x^2+1\right),\left(x^2+x+2\right),\left(x^2+2x+2\right)\right\}$

Also, for ${\mathrm{\mathbb{Z}}}_5\left[x\right]$, the monic irreducible polynomials are:

$$\begin{gathered} \left\{\left(x^2-2\right),\left(x^2+2\right),\left(x^2+2x-1\right), \\ \left(x^2-2x-1\right),\left(x^2-x+1\right), \\ \left(x^2+x+1\right),\left(x^2+2x-2\right), \\ \left(x^2-2x-2\right),\left(x^2+x-2\right), \\ \left(x^2-x-2\right)\right\} \end{gathered}$$

Hence, these are the required polynomials.