Remark 1

Let K be a field of characteristic $p \neq 0$ and L an extension of K. We have $\Omega_K(L) = 0$ if and only $L = K(L^p)$ (V, p. 103, Prop. 6). Therefore if L is finitely generated over K, then it is an algebraic and separable extension if and only if we have $L = K(L^p)$. When L is not finitely generated over K, this result no longer holds generally as is shown by the case where L is the perfect closure of K.