Let $p,q\in \mathbb{K}[X]$ be two polynomials and $\exists s\in \mathbb{K}[X]$ such that $q\cdot s=p$. Let $\mathbb{K}_p$ be the splitting field of $p$ and $\mathbb{K}_q$ be the splitting field of $q$. Is it always true that $\mathbb{K}_q\subseteq \mathbb{K}_p$?
Intuitively I would say it is true but unfortunately my algebra knowledge is very limited. I can't think of a counter-example.
It's "a" splitting field, not "the" splitting field. Such fields are unique only up to isomorphism. But it is true that any splitting field of $p$ contains a splitting field of $q$.