Corollary

— Let E be an extension of finite degree of K. For E to be separable it is necessary and sufficient that there exist a in E such that $\mathrm{Tr}_{E/K}(a) \neq 0$.

The condition is necessary by Prop. 1. Conversely, assume that there exists $a \in E$ such that $\mathrm{Tr}_{E/K}(a) \neq 0$. Given $x \neq 0$ in E, if we put $y = a x'$, then $\mathrm{Tr}_{E/K}(xy) \neq 0$. Now Prop. 1 shows that E is an etale algebra over K, hence a separable extension of K.