Let $E$ be finite separable extension of a field $k$. Let $K$ be the smallest normal extension of $k$ containing $E$. Does $K$ be separable?
Actually this is from the statements of Lang's the algebra, Corollary 1.6 at p.263., to show that $K$ is Galois extension. Lang states it obvious since $K$ is finite composite of finite number of conjugates of elements in $E$.
I showed that $K$ is such components, using Lang's remark on separable extension. But still have no idea to see $K$ is separable. By Lang's notation, $K = (\sigma_{1}E)(\sigma_{2}E)\cdots(\sigma_{n}E)$ where $n = [E:k]$, $\{ \sigma_{i} \}_{i=1}^{n}$ is embeddings $E \to k^{a}$, a algebraic closure of $k$ over $k$. Could you give me some hint for the reason why each $(\sigma_{i}E)$ is separable over $k$?