Degree of a splitting field is no greater than $n!$

Question

Is it possible to prove that if $P$ is a polynomial of degree $n$ then the degree of its splitting field is no greater than $n!$ without using notion of Galois group?

Answer 1

$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$Proceed by induction on $n$.

Let $\alpha$ be a root of $P$ in some extension field of the field $F$ you start with.

Then $\Size{F(\alpha) : F} \le n$, as $1, \alpha, \dots , \alpha^{n-1}$ span $F(\alpha)$ over $F$.

Moreover, since $\alpha$ is a root of $P$ in $F(\alpha)$, there is a polynomial $Q \in F(\alpha)[x]$ such that $$ P = (x - \alpha) Q. $$ Now $Q \in F(\alpha)[x]$ is a polynomial of degree $n-1$. Apply induction and use the multiplicativity formula for degrees.