Let $E\supset F$ be a finite Galois extension of fields. Show that there exists an element $a\in E$ such that $$\{\sigma(a):\sigma\in\mathrm{Gal}(E/F)\}$$ forms a basis of $E$ over $F$.
I guess that $a$ may be the primitive element of $E/F$ such that $E=F(a)$. Then $\sigma(a)$ are all the roots of the minimal polynomial of $a$. Only need to prove roots of this irreducible polynomial are linearly independent over $F$ and expand $E$. Am I in the right direction?