Let $L/K$ be an extension of fields, let $L_s$ be the separable closure of $K$ on $L$. Show that $L/L_s$ is purely inseparable and $L_s/ K$ is separable.
That $ L_s / K $ is separable, follows directly from the definition of $ L_s $. Could you help me show that $L /L_s$ is purely inseparable? Thanks..