Let $F$ $\subseteq$$E$$\subseteq$$K$ consecutive field extensions and $f$$\in$$F[X]$ a nonconstant polynomial.Prove that if $K$ is a splitting field of $f$ over $F$ ,then $K$ is a splitting field of $f$ over $E$.
What i did is: By the definition of splitting field $\exists$ $a_1,a_2,,,,a_m$ $\in$ $K$,$m=deg(f)$ such that $f(ai)=0$ $\forall$ $i=1,2,,,m$ and $K=F[a_1,,,a_m]$.
But $K=F[a_1,,,a_m]$ $\subseteq$ $E[a_1,,,,a_m]$ thus $E[a_1,,,,a_m]=K$ because $K$ contains $E$ as a subset and all the $a_i$'s.
We conclude that $K$ is a splitting field of $f$ over $E$
Is this proof correct or am i missing something?
Thank you!