Let $E/K$ be a field extension, and let $L_1$ and $L_2$ be intermediate fields of finite degree over $K$.
Prove that $[L_1L_2:K] = [L_1 : K][L_2 : K]$ implies $L_1\cap L_2 = K$.
My thinking process so far:
I've gotten that $K \subseteq L_1 \cap L_2$ because trivially both are intermediate fields over K.
I want to show that $L_1 \cap L_2 \subseteq K$, or equivalently that any element of $L_1 \cap L_2$ is also an element of $K$. So I suppose there exists some element $x\in L_1 \cap L_2\setminus K$. Well then I know that this element is algebraic over $K$, implying that $L_1:K=[L_1:K(x)][K(x):K]$, and similarly for $L_2$, implying that that these multiplied together equal $L_1L_2:K$ by hypothesis. And now I’m stuck in the mud... not knowing exactly where the contradiction is.