Let $L/K$ be an algebraic field extension. Denote by $K(T)= \mathrm{Frac}(K[T])$ the transcendental field extension of $K$. I would like to find out how to show that the equation $$K(T) \otimes_K L = L(T)$$ holds and especially where the requirement that $L$ is algebraic flows in.
That seems to be essential since for transcendent $L:= K(T)$ the formula above fails since $K(T) \otimes_K K(T)$ is not a field.
Thanks in advance!
by induction you must show that $K(T)\otimes K[\alpha]=K[\alpha](T)$ write $K[\alpha]=\frac{K[T]}{(f)}$ and prove the morphism $g\otimes (h+(f))\to g\pi(h)$ ($\pi$ is the natural morphism from $K[T]/(f)$ to L)is isomorphism.
(construct the converse by sending $\sum f_i(\alpha)T^i$ to $\sum T^i\otimes (f_i+(f))$)