order of the splitting field of $x^5 +x^4 +1 $? what  is the order of the splitting field of 
$x^5 +x^4 +1  = (x^2 +x +1)( x^3 +x+1)$  over $\mathbb{Z_2}$
i thinks  it will  $6$ because  $2.3 = 6$
Pliz help  me...

Because $x^5 +x^4 +1  = (x^2 +x +1)( x^3 +x+1)$ and the factors are irreducible, the splitting field of $x^5 +x^4 +1$ has degree at least $2 \cdot 3 = 6$.
On the other hand, the splitting field of $x^5 +x^4 +1$ is contained in $\mathbb F_{2^6}$, an extension of degree $6$. Hence this is the splitting field. Indeed, WA tells us that 
$$
x^{2^6}-x = x (x + 1) (x^2 + x + 1) (x^3 + x + 1) \cdots
$$
