I had a question I was stuck on:
Let $p(x)$ be a non-constant polynomial of degree $n$ in $F[x]$. Prove that there exists a splitting field $E$ for $p(x)$ such that $[E : F] \leq n!$.
My start:
By a theorem, we know that there exist a splitting field $E$ for $p(x)$ since $p(x) \in F[X]$.
$p(x)$ has degree $n$, and since $E$ of $F$ is a splitting field for $p(x)$, there are elements $\alpha_1 ..., \alpha_n$ in $E$ such that $E = F(\alpha_1 ..., \alpha_n)$ and also $p(x) = (x - \alpha_1)(x - \alpha_2)...(x - \alpha_n)$
I have no idea where to go from here, I feel as I'm really close. Any hints would be extremely helpful for me!
Hint: Use induction on the degree of the polynomial and multiplicativity of the dimensions in towers of fields.