Question

Find all irreducible polynomials of

(c) degree 2 in ${\mathrm{\mathbb{Z}}}_3\left[x\right]$

Short answer

The irreducible polynomial of degree 2 in ${\mathrm{\mathbb{Z}}}_3\left[x\right]$ are $x^2+1,x^2+x+2,x^2+2x+2,2x^2+2,2x^2+2x+1$, and $2x^2+x+1$.

Step-by-step solution

Irreducible polynomial

A polynomial is irreducible when its only divisors are its associates and nonzero constant polynomial.

The polynomial of degree 2 are in the form $a_2{x}^2+a_{1}x+a_0$ with $a_2\ne 0$ in ${\mathrm{\mathbb{Z}}}_3$and the elements in ${\mathrm{\mathbb{Z}}}_3$ are $\left\{0,1,2\right\}$.

Find Irreducible polynomial

Assume that $a_0=0$; then the polynomial is $x^2+a_{1}x=x\left(x+a_1\right)$, which is reducible. This implies that the only choice for $a_0$ is $a_0\ne 0$.

Find the monic polynomials as:

$$\begin{gathered} \left(x+1\right)\left(x+1\right)=x^2+2x+1 \\ \left(x+1\right)\left(x+2\right)=x^2+3x+2 \\ \left(x+2\right)\left(x+2\right)=x^2+4x+4 \end{gathered}$$

These monic polynomials are irreducible, so the left irreducible monic polynomial are$x^2+1,x^2+x+2$,and $x^2+2x+2$. When these irreducible monic polynomials are multiplied by 2, the obtained polynomials are also irreducible.

This implies that the irreducible polynomial of degree 2 in ${\mathrm{\mathbb{Z}}}_3\left[x\right]$ are $x^2+1,x^2+x+2,x^2+2x+2,2x^2+2,2x^2+2x+1$ and $2x^2+x+1$.