I'm thinking about the following claim:
Let $k^a$ be an algebraic closure of $k$, and let $K_i$ be a splitting field of $f_i$ in $k^a$ (the algebraic closure of $k$). Then the compositum of the $K_i$ is a splitting field for our family.
I wanted to prove this as follows. Let us remark that the compositum is defined as $$K:=k^a\left(\bigcup_{i\in I} K_i\right)$$ But then clearly $$f_i(X)=c\prod_{i\in I}(X-\alpha_i)$$ where $\alpha_i\in K$ by definiton and $K$ is generated by all the roots of all $f_i$. So $K$ is indeed a splitting field.
So now my first question is, does this work?
My second is why do we need $k^a$?
Thanks for your help