I read the proof of the Hasse Theorem at page 138 in "Arithmetic of elliptic curves" by J.Silverman and i don't understand why the Galois group of the extension $\overline{\mathbb{F}}_q/\mathbb{F}_q$ is generated by the $q^{th}$power map on $\overline{\mathbb{F}}_q$.

1$\begingroup$ This is a basic question about Galois theory of finite fields, I fear this is not suitable for this forum. $\endgroup$ – Chris Wuthrich May 22 at 16:45

$\begingroup$ @ChrisWuthrich $\overline{\mathbb{F}_p}$ is not a finite field $\endgroup$ – danihelovick May 23 at 22:18

$\begingroup$ This isn't researchlevel, in part because it's incorrect  the Galois group of the algebraic closure of a finite field is the profinite completion of $\mathbb{Z}$. See math.stackexchange.com/a/586134/364828 $\endgroup$ – user44191 May 23 at 23:47

$\begingroup$ @user44191 at page 138 it is said $\endgroup$ – danihelovick May 24 at 11:46

$\begingroup$ Doesn't it say "(topologically) generated" ? $\endgroup$ – Chris Wuthrich May 24 at 12:59