Exercise 11, § 15

Let K be a field of characteristic $p > 0$. For every commutative K-algebra A, raising to the p-th power makes the ring A into an A-algebra, hence also a K-algebra which is denoted by $A^p$^1. Let A be a commutative K-algebra. Show that the following conditions are equivalent : (i) A is a separable K-algebra. (ii) The unique K-homomorphism $\Phi : A \otimes_K K^{p^{-1}} \to A^{p^{-1}}$ such that $\Phi(a \otimes A) = a^p \cdot \lambda$ ($a \in A, \lambda \in K$) is injective. (iii) For every family $(b_i)_{i \in I}$ of K which is linearly free over $K^p$ and every family $(a_i)_{i \in I}$ of A the equality $\sum a_i^p b_i = 0$ implies that $a_i = 0$ for all i.

(It may be verified that (ii) $\Leftrightarrow$ (iii) and that (ii) is a consequence of V, p. 123, Th. 2, e).)