# On tensor product of field extensions

Let $$K$$ be a field which is a (transcendental) extension of $$\mathbb{C}$$. Let $$L_1, L_2$$ and $$M_1, M_2$$ be two field extensions of $$K$$ (not necessarily algebraic) such that $$L_1 \otimes_K L_2 \cong M_1 \otimes_K M_2$$ My question is: Is one of $$M_1$$ or $$M_2$$ a field extension (not necessarily finite) of $$L_1$$?

Let $$p_1,p_2,p_3,p_4$$ be primes, and let $$A_1,A_2,A_3,A_4$$ be extensions of $$K$$ of degrees $$p_1,p_2,p_3,p_4$$ respectively.
Let $$L_1 = A_1 \otimes_K A_2$$ $$L_2 = A_3 \otimes_K A_4$$ $$M_1 = A_1 \otimes_K A_3$$ $$M_2= A_2 \otimes_K A_4.$$
Then neither $$M_1$$ nor $$M_2$$ can be an extension of $$L_1$$ or $$L_2$$ because the degree of $$M_i$$ is never a multiple of the degree of $$L_j$$.