Extension E/K such that E/F is a splitting field The question asks us to prove that there is an extension $E/K$ such that $E/F$ is a splitting field of some polynomial $f(x) \in F[x]$ where $K/F$ is a finite extension. 
I'm not really sure how to go about doing this. I started by saying that $E$ is a splitting field where $p_i(x)$ is the irreducible polynomial of $\alpha_i$ over $F$. Because then we know $E$ over $K$ is a normal extension of $K$....
 A: As Eoin mentioned in a comment, your question is unclear because you did not clearly define what is a splitting field of a polynomial. If it is any field  extension where the polynomial splits, then the existence of a splitting field over $F$ has nothing to do with any particular field extension $K / F$, since existence gives absolutely no control over which of the many isomorphic splitting fields you may get. The question then is asking you to show that there is in fact one of them that also contains $K$. But if you already have the theorem that gives existence of splitting fields, then it is easy. Simply consider how your polynomial factorizes over $K$ and extend $K$ to $E$ to make it split completely. Then check whether $E$ is also a splitting field of the polynomial over $F$.
If, however, some algebraic closure of $F$ is fixed from the beginning, and "splitting field" is defined as the field obtained by adjoining all the roots of the polynomial in the algebraic closure, and all field extensions ever constructed are within the algebraic closure, then trivially $K$ must be within the splitting field.
