Exercise 29, § 1

Let k be a field of characteristic $\neq 2$, A the polynomial ring $k[T]$ in one indeterminate and $a, b$ two elements of k such that $a \neq 0, b \neq 0$ and $a \neq b$. Let $K = k(T)$ be the field of fractions of A and L, M the extensions of K adjoining to K respectively a root of $X^2 - T(T - a)$ and a root of $X^2 - T(T - b)$. Let B, C be the integral closures of A in L and M respectively, which are finitely generated free A-modules. Show that $L \otimes_K M$ is a field, a separable finite extension of K, but that $B \otimes_A C$, which is canonically identified with a subring of $L \otimes_K M$, is not the integral closure of A in $L \otimes_K M$.