Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal polynomial of $\alpha$. But then $L = \mathbb F_{p^{sd}} $. Since $\mathbb F_{p^{sd}}$ is unique, we see that this is the splitting field of every irreducible polynomial of degree $d$ over $K$.
Take $K = \mathbb F_2$ and let $P(X) = X^3 + X + 1$, $Q(X) = X^3 + X^2 + 1$. Let $L$ be the splitting field of $P$ and $L'$ be the splitting field of $Q$. The above tells us that $L$ and $L'$ are isomorphic. I would like to construct an explicit isomorphism between $L$ and $L'$.
I know that $L \cong \mathbb F_2[X] /(X^3 +X + 1)$ and $L' \cong \mathbb F_2[X] / (X^3 + X^2 + 1)$. Intuitively, I want to find an isomorphism $\phi : \mathbb F_2[X] \to \mathbb F_2[X]$ such that $\phi((X^3 + X + 1)) = (X^3 + X^2 + 1)$. A little playing around gives me $\phi(X) = X+1$. It now feels like I'm falling at the last hurdle: how do I finish the construction of an isomorphism between $L$ and $L'$? I don't think $\phi$ makes sense as a map from $L$ to $L'$, yet it seems the map I want.
Best Answer
You can think of $L$ as the set of all things of the form $ar^2+br+c$ where $a,b,c$ come from ${\bf F}_2$ and $r$ satisfies $r^3+r+1=0$. Now $L$ also contains a zero of $X^3+X^2+1$, so you are looking for the values of $a,b,c$ such that if $s=ar^2+br+c$ then $s^3+s^2+1=0$. You can just multiply everything out, use $r^3+r+1=0$ to get it down to a quadratic in $r$, set the coefficients to zero, and solve. This is probably a mess. There may be an easier way to do it, but this will get you an element of $L$ that satisfies $X^3+X^2+1=0$. Once you have that element, you know the isomorphism you're looking for takes $r$ to that element. Since $r$ generates $L$, you get the entire isomorphism.
Now in fact I think you have managed to find that (in my notation) $r+1$ is the element you are looking for. So your map from $L$ to $L'$ takes $r+1$ in $L$ to a generator, call it $t$, in $L'$. Might be easier to see it as a map from $L'$ to $L$, taking $at^2+bt+c$ to $a(r+1)^2+b(r+1)+c$.