Galois group of an arbitrary field extension

Question

It is well known that if $E/F$ is a finite field extension, then $|Gal(E/F)|\leq [E:F]$. One can also prove this for extension with $|Gal(E/F)|$ is countable, by Theorem 13, Page 36 of [E. Artin, Galois Theory]. Now, if $E/F$ is an arbitrary field extension with $|Gal(E/F)|=\alpha$, where $\alpha$ is an arbitrary cardinal number, is it true that $[E:F]\geq \alpha$? (i.e., $v.dim_F(E)\geq \alpha$?).

Answer 1

This is not true in general, even if $E/F$ is algebraic and Galois. Consider $\overline{\Bbb F_p}/\Bbb F_p$. Here $\mathrm{Gal}(\overline{\Bbb F_p}/\Bbb F_p)$ is uncountable, but $[\overline{\Bbb F_p}:\Bbb F_p]$ is countable.