Consider $A$ and $B$ two intermediate fields of the field extension F/K.
I have already proved that $[AB:K]\leq {[A:K][B:K]\over [A\cap B: K]}$.
I would like to find a simple example (for example, taking K= $\mathbb Q)$ in which it doesn't hold the equality (a counterexaple to show that the equality only happens sometimes) , but all the examples that come to my mand follow the equality.
Do you have any idea?
Thank you so much!
Let $K = \mathbb{Q}$, $A = \mathbb{Q}(2^{1/3})$ and $B = \mathbb{Q}(\zeta2^{1/3})$ (here $\zeta$ denotes a primitive cube root of one, so that $\zeta2^{1/3}$ is one of the other roots of the polynomial $x^3 - 2$).
The right hand side of your inequality is nine since these fields obviously intersect to $\mathbb{Q}$ (being unequal extensions of degree 3). The left hand side is six because the compositum of $A$ and $B$ is the splitting field of $x^3 - 2$, which splits in any field containing $2^{1/3}$ and $\zeta$, but the minimal polynomial of $\zeta$ over $\mathbb{Q}$ is just $x^2 + x + 1$.