Finding irreducible polynomial in finite field

Question

I would like to find an irreducible polynomial of degree $3$ in $\mathbb{F}_4$, where $$\mathbb{F}_4 = \{a+b\alpha| \ a, b\in \mathbb{F}_2, \alpha^2 = \alpha + 1\}.$$ I first tried to find an irreducible polynomial of degree 2. Since $\mathbb{F}_4 = \mathbb{F_2[\alpha]}$, we know $f(x) = x^2 - x - 1$ is irreducible since $f(\alpha) = 0$ and its degree matches the degree of the simple extension. However, when it comes to finding a degree 3 irreducible polynomial, I feel it would be very difficult to argue whether a given polynomial is irreducible. Any suggestions on how to approach this?

Answer 1

Deciding whether a degree $3$ polynomial is irreducible over $\mathbb F_4$ is actually quite easy. If a degree $3$ polynomial factors then at least one of those factors must be degree $1$, i.e., your polynomial must have a root. So a degree $3$ polynomial factors if and only if it has a root. As your field has only $4$ elements this is straightforward to check.