Minimum splitting field of the polynomial over GF(2)

Question

I need to find the minimum splitting field of the polynomial over GF(2): $$x^5+x^4+1.$$ I find that $x^5+x^4+1 = (x + \alpha)^2(x + \alpha + 1)(x + \alpha^2)(x + \alpha^2 + \alpha)$ over GF(8). But I don't know if it's right, because $x =\alpha $ is a multiple root. Am I right? And what is right? Help me please!

Answer 1

$x^5+x^4+1=(x^2 + x + 1) (x^3 + x + 1)$ in $GF(2)$.

These factors are irreducible because they have no root in $GF(2)$.

The splitting field of $x^2 + x + 1$ has degree $2$ over $GF(2)$.

The splitting field of $x^3 + x + 1$ has degree $3$ over $GF(2)$.

Therefore, the splitting field of $x^5+x^4+1$ has degree $6$ over $GF(2)$.