The monic quadratic polynomials in $\mathbb Z_2[x]$ are -
$x^2, x^2 + 1, x^2 + x, x^2 + x + 1$
$x^2 = x \cdot x$ so is reducible
$x^2 + 1 = (x + 1) \cdot (x + 1)$ so is reducible
$x^2 + x = x \cdot (x + 1)$ so is reducible
$x^2 + x + 1$ - The fact that this polynomial has no root in $\mathbb Z_2[x] $, that $\mathbb Z_2$ is a field, and that this polynomial has degree $\le 3$ implies that this polynomial is irreducible.
Have I got all that correct? In particular is the last claim correct or have I made an incorrect implication?