Let $f(x)\in F[x]$ be a polynomial of degree $n$. Let $K$ be a splitting field of $f(x)$ over $F$. Then [K:F] must divides $n!$.
I had this mis-concept, since $f(x)$ can have at most n roots. then its splitting field degree can at most be $n$
to make this more intuitive, take cubic polynomial for example, can someone give me an example of cubic polynomial which has splitting filed degree equal to 6. does this mean this polynomial now have 6 roots, as splitting filed in my understanding is $Q(\alpha_1, \alpha_2,\alpha_3)$