Exercise 8, § 3

Let $k$ be a field and $K$ an extension of $k$. One says that $K$ is formally étale over $k$ if, for every $k$-algebra $A$, every ideal $a$ of $A$, square zero, and every $k$-homomorphism $u$ of $K$ into $A/a$, there exists a unique $k$-homomorphism $v$ of $K$ into $A$ which yields $u$ by passing to the quotient. a) If $K$ is formally étale over $k$, the equivalent conditions of the preceding exercise are satisfied. (If $M$ is a $K$-module, consider the $k$-algebra whose underlying $k$-vector space is $K \oplus M$, $M$ being an ideal of square zero in it.) b) Conversely, if the equivalent conditions of the preceding exercise are satisfied, then $K$ is formally étale over $k$. (Since the field $K$ is separable over $k$, Exercise 4 makes it possible to prove the existence of $v$.) c) Let $B = (b_i)_{i \in I}$ be a family of elements of $K$ such that $(db_i)_{i \in I}$ is a basis of $\Omega_k(K)$ over $K$. If $K$ is separable over $k$, then $k(B)$ is a purely transcendental extension of $k$ (use Theorem 2 of A, V, p. 125, Theorem 1 of A, V, p. 97 and Exercise 6 of A, V, p. 165) and $K$ is formally étale over $k(B)$.