Exercise 19, § 14

Let E and F be two transcendental extensions of a field K, linearly disjoint over K. Show that K(E U F) is distinct from the ring C (isomorphic to $E \otimes_K F$) generated by E U F. (Reduce to the case where the transcendence degrees of E and F over K are equal to 1 ; if $x \in E$ and $y \in F$ are transcendental over K show that we cannot have $(x + y)^{-1} \in C$; arguing by contradiction, show that in the opposite case, if p is the characteristic exponent of K, there would exist an integer $r \geq 0$ such that $(x + y)^{-p^r}$ belonged to the subring of C generated by $K(x) \cup K(y)$.)