Proposition 6

— Let K be a field (of characteristic p) and L an extension of K. a) Let $x \in L$; the differential $d_{L/K}x$ vanishes if and only if x belongs to $K(L^p)$. b) $\Omega_K(L) = 0$ if and only if $L = K(L^p)$.* c) Suppose that L is p-radical of finite height over K; then $\Omega_K(L) = 0$ if and only if $L = K$.