Exercise 7, § 3

Let k be a field, $P$ its prime subfield, $K$ an extension of $k$. Then the following conditions are equivalent: a) Every derivation of $k$ into a $K$-module $M$ extends uniquely to a derivation of $K$ into $M$. b) One has $\Omega_P(K) = \Omega_P(k) \otimes_k K$. c) The field $K$ is separable over $k$ and $\Omega_k(K) = 0$. If $k$ is of characteristic 0, these conditions are also equivalent to the condition: d) The field $K$ is an algebraic extension of $k$. If $k$ is of nonzero characteristic $p$, they are also equivalent to the conditions: e) $K = k \otimes_{k^p} K^p$. f) Every p-basis of $k$ is also a p-basis of $K$.