Question about separable extension

Question

Let $K/F$ be a field extension such that $K=F( u_1, u_2,...,u_n) $.

How do I prove that $K/F$ is separable if and only if each $u_i$ is separable over $F$?

Answer 1

One way is obvious. Now use induction on n. For the other side use the fact $K/F$ is separable iff # of $F$-isomorphisms of $K$ into $L =[K:F]$ where $L$ is some normal extension of $F$ containing $K$. So we have #$F$ isomorphisms of $F(u_i)\rightarrow L=F(u_1):F$ By induction $K/F(u_1)$ is separable and hence #$F(u_1)$ isomorphisms of $K\rightarrow L=K:f(u_1)$ So #$(F$ isomorphisms of $K\rightarrow L)= $ #$(F$ isomorphisms of $F(u_i)\rightarrow L).$#$(F(u_1)$ isomorphisms of $K\rightarrow L)$$=[F(u_1):F].[K:F(u_1)]=K:F$. Hence $K/F$ separable.