I was reading Aluffi's book algebra chapter 0 , in theorem 6.9 he gives us six equivalent condition to define (finite) galois extension , and i'm stuck in one implication:
If $F$ is a finite extension of $k$ then $|Aut_k(F)|=[F：k]$ implies that $F$ is normal and separable extension of $k$
The textbook told me to use this theorem:
Let $F$ be a finite and separable extension of $k$ then$|Aut_k(F)|\le [F：k]$ with equality holds iff $F$ is a normal extension of$k$.
But i can only use this theorem to prove the-other-direction implication . How can i prove the above question? Thanks.