Intersection of normal extensions is a normal extension

Question

Let's assume that we have two normal extensions $L_1$ and $L_2$ which means that $L_1=K(A_1)$ and $L_2=K(A_2)$, where $A_1$ and $A_2$ are sets of roots of proper polynomials. How can I show that intersection of $L_1$ and $L_2$ is also normal extension? Is it clear because of the fact that $L_1 \cap L_2=K(A_1 \cap A_2) $? I don't know if it's sufficient for the proof.

Answer 1

No, it is not even correct that $L_1\cap L_2=K(A_1\cap A_2)$.
You can prove the claim as follows: Take any irreducible Polynomial in $K[t]$ that has a root in $L_1\cap L_2$. By normality of $L_1,L_2$ it will split into linear factors over $L_1,L_2$. By unique factorization these factorizations coincide and are therefore already in $L_1\cap L_2$.