Exercise 12, § 2

Let K be a commutative field, E and F extensions of K, and A the ring $E\otimes_KF$.

a) If E and F are transcendental extensions of K, then the ring A is not local.

b) Suppose that E is an algebraic extension of K, and denote the separable closures of K in E and F, respectively, by $E_s$ and $F_s$. The following properties are equivalent:

(i) The ring A is local.

(ii) The ring $E\otimes_KF_s$ is an integral domain.

(iii) The ring $E\otimes_KF_s$ is a field.

(iv) The ring $E_s\otimes_KF_s$ is a field.

When they hold, the maximal ideal $\mathfrak{m}$ of A is the set of nilpotent elements of A, and every composite extension of E and F is isomorphic to $A/\mathfrak{m}$.