$E/K$ is finite Galois extension. Let $G(E/K)=\{\sigma_i\}$ be its Galois group. Then $\sigma_i$ are linearly independent over $E$?

Question

$E/K$ is finite Galois extension. Let $G(E/K)=\{\sigma_i\}$ be its Galois group.

$\textbf{Q1}:$ Then $\sigma_i$ are linearly independent over $E$?(I do not want to use normal basis statement to deduce $E\otimes_KE\cong E[G]$ and this will assert linear independence over $E$). How do I see this directly?

$\textbf{Q2}:$ This is a step used in the proof of Hilbert 90 thm. However, I do not see I need cyclicness of the group for above question. Am I right on this?