Corollary 2

— *Let $x$ be an element of $\Omega$ of degree $n$ over $K$. For $x$ to be separable over $K$ it is necessary and sufficient that $x$ should have $n$ conjugates in $\Omega$; when this is so, all the conjugates of $x$ over $K$ are separable over $K$.

Let $f$ be the minimal polynomial of $x$ over $K$; its roots are the conjugates of $x$ over $K$, and each of these roots admits $f$ as minimal polynomial over $K$. Now $x$ is separable over $K$ if and only if the polynomial $f$ is separable ($V$, p. 39, Prop. 5), that is, if $f$ has $n$ distinct roots in $\Omega$. Cor. 2 follows from this.*