Prove if $\deg(f)=d$ and $L$ is the splitting field of $f$ over $K$, then $[L:K]\mid d!$

Question

I've been a lot of time trying to figure how to prove this statement from my Galois Theory course:

Let $L$ be the splitting field over $K$ of $p(X)\in K[X]$ a polynomial of degree $d$, then $[L:K]\mid d!$.

My course notes give me a hint, it says to study two different cases when applying induction hypothesis wether $p$ is irreducible or not.

I guess my induction initial case must be $deg(p)=1$ or $deg(p)=2$, since in both cases te statement is obviously true, but I really don't know to translate the divisible condition when adding $1$ to the degree of $p$. I'm sorry if this is too trivial and I'm just not getting something really easy. I've seen other similar posts in this site, but they ask to prove that $[L:K]$ is less or equal to $d!$, instead of proving divisibility. How can I solve this? Any help or hint will be appreciated, thanks in advance.

Answer 1

The induction goes like this:

If $p$ has degree $1$ or $2$, then the result is obvious, as you said. Suppose that the claim is true for any $m<n$. Let us prove it for $n$. So pick some polynomial $p \in K[X]$ such that $\deg(p) = n$. We have two cases:
$\textbf{1:}$ $p$ is reducible. Then $p = p_1p_2$, with $p_1, p_2 \in K[X]$ nonconstant polynomials. Suppose that $\deg(p_1) = e$. Then $\deg(p_2) = n - e$. Take $L_1$ the splitting field over $K$ of $p_1$. Then the splitting field over $K$ of $p$, which we denote by $L$ will be the splitting field of $p_2$ over $L_1$. By our induction hypothesis, $[L_1:K] \mid e!$ and $[L:L_1] \mid (n-e)!$. Therefore, $[L:K] \mid e! (n-e)!$ which is a divisor of $n!$
$\textbf{2:}$ $p$ is irreducible. Take $\alpha$ a root of $p$. Then $[K(\alpha):K] = n$. Over the field $K(\alpha)$ we can write $p = (X- \alpha)q$, where $\deg(q) \le (n-1)$. Again, the field $L$ will be the splitting field over $K(\alpha)$ of $q$. By our induction hypothesis, $[L:K(\alpha)] \mid (n-1)!$. Consequently, we obtain that the degree $[L:K] = [L:K(\alpha)] \cdot [K(\alpha):K]$ is a divisor of $(n-1)! \cdot n = n!$