# prove that $K$ is the composite field of $K_i$'s

Let $$\{f_i\}_{i \in I}$$ be a family of polynomials in $$F[X]$$ (F is of course some field). Consider the extension field $$K$$ such that $$f_i$$ splits in $$K[X]$$ and is generated by all the roots of $$f_i,i\in I$$ . Let $$\bar{F}$$ be an algebraic closure of $$F$$ and $$K_i$$ denote the splitting field for the family $$f_i$$ in $$\bar{F}$$. Then prove that $$K$$ is the composite field of $$K_i$$'s .