I am self studying Field Theory from Textbook Hungerford and couldn't solve this problem on page 241 .
For (a) X would be composite of LM if X = intersection of all subfields P such that $L\bigcap M\subseteq P$.
So, I tried using this defination to prove $LM \subseteq L(S) $ and $L(S) \subseteq LM$.
I tried proving $LM \subseteq L(S) $but I got confused by inclusion relations as K is subset of $ L \bigcap M $ and then M= K(S) for some S subset of M.
It's a very hard nut to crack due to it and I am unable to.
I have done (b).
For (c) I have no clue. (a) can't be used in C as $E_i$ can't be necessarily subset of each other,in my opinion.
So, can you please tell how to solve (a) and Give hint for (c).
Part a), your definition of the composition $LM$ seems to be wrong. The definition of $LM$ is the the smallest subfield containing both $L$ and $M$, not the intersection $L\cap M$.
First note that $L(S) \supset K(S) =M$ (because $L\supset K$). Clearly $L\subset L(S)$. So $LM \subset L(S)$. On the other hand, $L(S) \subset (LM)(S) = LM$ since $L\subset LM$ and $S\subset M$.
Part c), you can prove inductively on $n$. (Try prove the case for $n=2$ first.)