Let $(L_i)_{i \in I}$ be an indexed Family of normal field extensions over $K$ ($L_i \subseteq \overline{K}$ for an algebraic closure of $K$). Show that $\bigcap_{i\in I} L_i =: L$ is also a normal extension over $K$.
I'm not sure if my proof is correct so any criticism is very welcomed:
Let $f \in K[X]$ be an irreducible polynomial and $\alpha \in L$ be a root. Then $\alpha \in L_i$ for every $i\in I$ and by assumption, $f$ splits into linear factors in $L_i[X]$ for every $i \in I$. In $\overline{K}[X]$ we have that: $$f = (X-\alpha_{i_1})\cdot ... \cdot (X-\alpha_{i_n}) = (X-\alpha_{j_1})\cdot ...\cdot (X-\alpha_{j_n})$$ for every $i \neq j \in I$. Since $\overline{K}[X]$ is a UFD, the factorization on the RHS is just a reordering of the LHS, hence $\alpha_{i_k} = \alpha_{j_l}$. So $f$ splits into linear factors in $\bigcap_{i\in I}(L_i[X]) = (\bigcap_{i \in I}L_i)[X] = L[X]$.
Specifically, I'm not sure if my last step was correct. Any advice greatly appreciated!