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 $L\_1$?



On tensor product of field extensions
Not necessarily.


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 $L\_j$.


