Exercise 12, § 15

Let K be a field and $(X_i)_{i \in I}$, a family of indeterminates. Show that the algebra of formal power series $K\{[(X_i)_{i \in I}]\}$ is a separable K-algebra. (Apply Ex. 11 ; it may also be verified that for every finite extension K' of K, $K\{[(X_i)_{i \in I}]\} \otimes_K K'$ is isomorphic to $K'\{[(X_i)_{i \in I}]\}$ and then Th. 2, d) of V, p. 123 applied.) Deduce that the field of fractions of $K\{[(X_i)_{i \in I}]\}$ is a separable extension of K (V, p. 120, Prop. 4).