Show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$

Question

How do I show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$? Someone says I should use the fact that the range of the matrix is 7, but I don't exactly know how that applies. Thanks for any input.

Answer 1

Hint: Note it has no roots. List all the irreducibles of degree 4 or less. Division.

With degree 4, it's not enough to test for absence of roots.