$F/K$ field extension, prove if $|E_1:K|$, $|E_2:K|$ are coprime numbers, then $E_1 \cap E_2 = K$

Question

Let $F/K$ be a field extension, and let $E_1$, $E_2$ be two fields that are between $K$ and $F$. I want to prove that if $|E_1:K|$ and $|E_2:K|$ are coprime numbers, then $E_1 \cap E_2 = K$.
I know that if the degree of a field extension $A/B$ is prime then there are no fields between $A$ and $B$. I've tried to work with that but found nothing so far.
Also the fact that I don't know if $F/K$ is finite is causing me a lot of trouble.
Could someone help me?

Answer 1

Hint

$[E_1:K] = [E_1:E_1 \cap E_2][E_1 \cap E_2:K]$ and $[E_2:K] = [E_2:E_1 \cap E_2][E_1 \cap E_2:K]$ as the intersection of two fields is a field.

Derive a contradiction if $[E_1 \cap E_2:K] \neq 1$.