Let $K$ be a splitting field of $f(x)$ over $F$. If $E$ is a field such that $F\subseteq E\subseteq K$, show that $K$ is a splitting field of $f(x)$ over $E$.
We know that $$f(x) = c(x-u_1)(x-u_2)\dots(x-u_n),$$ where $c \in F \subseteq E \implies c \in E$.
Also we know that $$K=F(u_1, u_2, \dots, u_n).$$
Do we need to show that $K=E(u_1,u_2,\dots,u_n)$? If so, how can we do that?
Or am I completely on the wrong track?
Note that if $K = F(u_1,u_2,\ldots,u_n)$ means $K$ is "the smallest subfield of $K$ containing $u_1,u_2,\ldots,u_n$ and $F$. Now $E(u_1,u_2,\ldots,u_n)$ is "the smallest subfield of $K$ containing $(u_1,u_2,\ldots,u_n)$ and $E$. But since $E \subset K$, there is no difference between $E(u_1,u_2,\ldots,u_n)$ and $F(u_1,u_2,\ldots,u_n)$, so $K = (u_1,u_2,\ldots,u_n)$.
When you say "$K$ is the splitting field of $f(x)$ over $F$", just think "$K$ is the smallest field with $F$ plus the roots of $f$". So if you change the base field to any intermediate field between $E$ and $K$, it makes no difference.