I am interested in the splitting field $L$ of $f=X^4+X^3+1\in\mathbb{Q}[X]$.
As we can see here, we have $Gal(f)\cong S_4$ by using Dedekind's Theorem (Question 1: Can we obtain this result without using this theorem?). So $[L:\mathbb Q]=24$. We can also see that $f$ doesn't have real roots so it factors to $f=(X-\alpha)(X-\overline{\alpha})(X-\beta)(X-\overline{\beta})$. Therefore we should be able to conclude $L=\mathbb{Q}(\alpha,\beta)$. Now there must be an error in my train of thought since this field can at most have degree $4\cdot 4$ over $\mathbb Q$.
Question 2: What is the correct splitting field?