A doubt on generalizing a result on Extension fields

Question

I had a doubt in generalizing a result on extension of fields involving composites. Upon searching, I came across this Extension Degree of Fields Composite.
We have a result that, if $L$ and $M$ are two finite extensions of a field $F,$ then $$[LM:F] \leq [L:F][M:F]$$ with the equality if and only if an $F$-basis for one of the fields remains linearly independent over the other.

Then I thought of writing that result as follows without the use of inequality. $$[LM:F]=\frac{[L:F][M:F]}{[L \cap M:F]}$$ I am not sure whether this is true or not.

The doubt is about the equality in the above result.

My attempt: $F$-basis for $L/F$ will have $1$ in it. Similarly, $F$-basis for $M/F$ will have $1$ in it. In their intersection, we will have $1.$ I am not able to go ahead.

In the case of the linked question, it is stated that the converse given there holds good when $[L:F]=2$ or $[M:F]=2.$ I could not get why this is true.

Any ideas on how to progress will be of great help.

Answer 1

We may as well assume that $L\cap M=F$, so the question is whether if $L\cap M=F$ then $[LM:F]=[L:F][M:F]$. Translating this into group theory via the Galois correspondence, this is asking whether if $H$ and $K$ are subgroups of a finite group $G$ which together generate $G$, then $[G:H\cap K]=[G:H][G:K]$. This is not true: for instance, if $G=S_3$ then you could take $H$ and $K$ to be the subgroups generated by two different $2$-cycles, and then $[G:H\cap K]=6\neq 9=[G:H][G:K]$.

In terms of fields, if you take a Galois extension $E/F$ with Galois group $S_3$, it will have three different subfields of degree $3$ over $F$ (corresponding to the three subgroups of $S_3$ of order $2$). If you take $L$ and $M$ to be two of these subfields, then $LM=E$ and $L\cap M=F$, but $[E:F]=6\neq 9=[L:F][M:F]$.

Very explicitly, for instance, you could take $F=\mathbb{Q}$, $L=\mathbb{Q}(\sqrt[3]{2})$, and $M=\mathbb{Q}(e^{2\pi i/3}\sqrt[3]{2})$. Then $[L:F]=[M:F]=3$ but $[LM:F]=6$. ($L$ and $M$ are formed by adjoining two different roots of $x^3-2$ and $LM$ is the splitting field of $x^3-2$.)