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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 .

I could not come up with anything. Can someone please help to prove this fact.

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