Finding All Irreducible Polynomials in $P_3(\mathbb{F_3})$

Question

Suppose I want to find all irreducible polynomials in $P_3(\mathbb{F_3})$.

From what I understand, a polynomial is not irreducible if it can be written as a product of two simpler polynomials where those degrees are smaller than the one I started with.

$P_3=a_3x^3+a_2x^2+a_1x+a_0$
$\mathbb{F_3}=\{1,2,3\}$

Now I'm a bit confused as to how I'm suppose to do this. There are quite a bit of polynomials, and even if I do find all possible polynomials, I'm not sure how to test if they are irreducible or not.

Any help would be appreciated. I believe there may be a trick to this to speed things up.

Thanks.

Answer 1

From the Factor Theorem (for $f(x) \in F[x]$, $F$ a field, $a \in F$ is a root of $f(x)$ iff $x-a$ is a factor of $f(x)$ in $F[x]$), we can derive a corollary that states that if $f(x)$ is a polynomial of degree 2 or 3, then $f(x)$ is irreducible in $F[x]$ iff $f(x)$ has no roots in $F$.

So, in short, you just need to check for each polynomial if it has any roots in $\mathbb{F}_3$.