Exercise 6, § 3

Let k be a field and K an extension of k. Suppose that K is a smooth k-algebra. Let $k'$ be an algebraic extension of finite degree of k. a) The ring $K \otimes_k k'$ is a product of a finite number of local Artinian rings, which are smooth $k'$-algebras (use Exercise 2, p. 76). b) The ring $K \otimes_k k'$ is reduced (use the preceding exercise). c) The field K is separable over k.