A nonempty intersection of normal subextensions inside a common algebraic extension is normal

The intersection $K$ is an intermediate field of $M\mathrm{/}F$. Since $M\mathrm{/}F$ is algebraic, every element of $K$ is algebraic over $F$.

Fix $\alpha \in K$ and let $m\in F[x]$ be its minimal polynomial from [F2]. For every $i\in I$, one has $\alpha \in E_i$, so [F1] makes $m$ split over $E_i$.

Because $I$ is nonempty, choose $i_0\in I$ and write the linear factorisation of $m$ in $E_{i_0}[x]\subseteq M[x]$. For any $i\in I$, a linear factorisation also exists in $E_i[x]\subseteq M[x]$. Uniqueness of factorisation in the ring $M[x]$ from [F3] shows that the same roots, with the same multiplicities, occur in both factorizations. Hence every root from the first factorisation lies in every $E_i$, and therefore in $K$.

Thus the minimal polynomial of every $\alpha \in K$ splits over $K$. Together with algebraicity from step 1.1, [F1] shows that $K\mathrm{/}F$ is normal. The nonempty hypothesis was used in step 2.1; without it the intersection convention could give the ambient $M$, which need not be normal.