For Galois extension $L:K$, does $L = K(\alpha)$ imply $\{\sigma_1(\alpha), \dots, \sigma_n(\alpha)\}$ is a basis for $L$ over $K$

Question

For Galois extension $L:K$ with Galois group $\{\sigma_1, \dots, \sigma_n\}$, does $L = K(\alpha)$ imply $\{\sigma_1(\alpha), \dots, \sigma_n(\alpha)\}$ is a basis for $L$ over $K$?

The proof that I've seen for the normal basis theorem starts with a primitive element $\alpha \in L$ and then switches to another element $\beta \in L$ to show $\{\sigma_1(\beta), \dots, \sigma_n(\beta)\}$ is a basis for $L$ over $K$.

Does the result still hold for $\alpha \in L$?

Answer 1

Consider $L=\mathbb{C}$, $K=\mathbb{R}$. Then $\mathbb{C}=\mathbb{R}(i)$, but $i,-i$ is not a basis for $\mathbb{C}$ as $\mathbb{R}$-vector space. Generally, not any primitive element will do.