$[E:F]$ and $[E':F']$ are coprime $\Longrightarrow [EE':F] = [E:F][E':F]$.

Question

Let $E$ and $E'$ be intermediate fields of a field extension $K/F$ such that $[E:F]$ and $[E':F']$ are coprime. Show that $[EE':F] = [E:F][E':F]$.

Obiously, we have $[EE':F] = [EE':E][E:F] \leqslant [E':F][E:F]$, but how to prove they are qaual? Thanks in advance.:-)

Answer 1

Well, by your above reasoning, you have that $[E:F]$ divides $[EE':F]$. Similarly, $[E':F]$ also divides $[EE':F]$. This implies $lcm([E:F],[E':F])$ divides $[EE':F].$ Since they are assumed co-prime, $lcm([E:F],[E':F])=[E:F][E':F].$

So in conclusion, for some positive $k\in \mathbb{N}$, $$ k[E:F][E':F]=[EE':F]\leq [E:F][E':F], $$ implying that $k=1$, yielding the desired.