Let $F_2$ be the field with 2 elements.
(a) Factor the polynomial $ f(x) = x^4+x^2+x+1 \in F_2[x]$
into irreducible factors in the ring $F_2[x]$.
(b) Let $E$ be the splitting field of f over $F_2$. How many elements does $E$ have?
I calculated the solution for (a) as
$x^4+x^2+x+1 = (x^3+x^2+x+1)*(x+1)$
For (b) I figured out that (x+1) is redundant because it already splits in $F_2$
I want to find the splitting field $E$ explicitly. I think it is given by $F_{2^3}$ but I don't know how to prove that. Can someone help me?
I think the splitting field of an irreducible polynomial $f \in F_p[X]$ with $deg(f) = n$ is always isomorphic to $F_{p^n}$ but I can't find that statement anywhere. Is it true?