Let $L,K,L_1,L_2$ be fields where $L/K$ is a field extension and $K \leq L_1,L_2 \leq L$ be intermediate fields. We define $L_1L_2$ to be the subfield of $L$ generated by $L_1 \cup L_2$
Show the following: $L_1L_2=L_1(L_2)=L_2(L_1)$
My thoughts: I am a little bit confused what the expression $L_1(L_2)$ means. I have not seen $()$ being defined for fields, nor what the multiplication of fields is. The thing that comes to mind is: Let R be a ring. For $S \subset R$
$(S):=\bigcap \{A: A$ is an Ideal of $R$ with $S \subseteq A\}$
In this case does $L_1L_2$ mean the multiplication of ideals which would be $L_1L_2:=\{a_1b_1+...+a_nb_n:a \in L_1, b \in L_2\}$
Is the meaning of the notation the same for fields as for rings?
If you have a field $L$ and a subfield $F \subseteq L$ and a subset $A \subseteq L$, then $F (A)$ means the smallest subfield of $L$ containing both $F$ and $A$. If you have subfields $F_1$ and $F_2$ then $F_1 F_2$ is the compositum, which is the smallest subfield of $L$ containing both $F_1$ and $F_2$.