I'm working with $GF(4)$ and I'm looking for irreducible polynomials of different degrees over that field. So $GF(4) = \{0,1,\alpha, \alpha + 1\}$ where $\alpha^{2} + \alpha + 1 = 0$, and first I'm looking for an irreducible polynomial of degree 2.

I looked at $\alpha x^2 + x + 1$, and it seems to be irreducible, but in the suggested solution to the problem they have used $x^2 + \alpha x + 1$, which is clearly irreducible. They suggests monic polynomials for the rest of the problems as well, and I'm wondering if they need to be monic?