Corollary 1

— Let $L$ be a finitely generated extension of a field $K$ of transcendence degree $s$. The vector space $\Omega_K(L)$ is of dimension $\geq s$ over $L$, with equality if and only if $L$ is separable over $K$.

Let P be the prime subfield of K. Let N be the kernel of α, then by exactness of the sequence $(\mathbf{E}_{P,K,L})$ (V, p. 128) and Th. 4, we have $[ \Omega_K(L) : L ] = s + [N : L]$; by V, p. 132, Cor., the extension L of K is separable if and only if $N = 0$. The corollary now follows.