I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields.
My question is as follows: Let E and F be algebraically closed fields containing an algebraically closed field K of characteristic 0. Suppose E is algebraically independent from F over K, and E and F are contained a common field.
Let E' and F' be the algebraic closures of the fields E(t) and F(t) of rational functions over E and F respectively.
Then, is the intersection of $E'\cdot F'$ and the compostie field EF equal to $E\cdot F$? that is, $$E'\cdot F'\cap EF=E\cdot F\ ?$$
where for fields M and N, $M\cdot N:=\{x\cdot y:x\in M,\ y\in N\}$ so that $M\cdot N\setminus\{0\}$ is an abelian group with respect to the multiplication.
At least, we can check that $$E(t)\cdot F(t)\cap EF=E\cdot F$$ because of the unique factorization in the polynomial ring over a field.
