If one has an extension of domains $D_1\subset D_2$ such that the extension of fields of fractions $K_{D_2}/K_{D_1}$ is transcendental of degree $r$ (with $K_{D_i}$ the field of fractions of $D_i$), is it always possible to find $r$ algebraically independent elements over $K_{D_1}$ that lie in $D_2$?
Obviously you can take $r$ algebraically independent elements in $D_2$; you can even take them with the same denominator; let's say $u_1/v,\ldots,u_r/v$. Now, I haven't been able to prove if the $u_i$ are algebraically independent over $D_1$...