let $E$ be a splitting field of $p(x)$ the minimal polynomial of $u$ over $K$ then $E \subset F$ where $F=K(u)$ is normal over K, $u \in F$

Question

I have a hard time struggling to understand this: Let $u \in F $ be algebraic over $K$. If $F=K(u)$ is normal over $K$, let $E$ be a splitting field of $p(x)$ the minimal polynomial of $u$ over $K$ then $E \subset F$ (then obviously implies that $E=F$)

How is $E=K(u,u_1, \ldots , u_n)$ a subset of $F=K(u)$ ?

Answer 1

If $F/K$ is normal then since it contains the root $u$ of the irreducible polynomial $p(x)$, it must contain all roots of $p(x)$, i.e. $p(x)$ splits over $F$. Since $E$ is the splitting field of $p(x)$ we must have $E \subset F$ (or really $F$ must contain an isomorphic copy of $E$).