is galois group always finite?

Question

I am new to Galois theory. my problem is

let F be an extension field of a field K the cardinality of Galois group denote by $|Aut_KF|$ is always finite?

I try to find a contradiction on $\Bbb R$

but, since $\Bbb Q$ is the minimal sub field and $|Aut_{\Bbb Q} \Bbb F|=1$ it is impossible on $\Bbb R$

is above statement true or are there any example that contradict it

Answer 1

The order of the Galois group is precisely the degree of the field extension $[F:K]$, so one needs to examine an infinite extension to get an infinite Galois group. For instance, if $\overline{\mathbb{F}_p}$ is an algebraic closure of $\mathbb{F}_p$, then $$ \text{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p)\cong \hat{\mathbb{Z}} $$ which is a profinite group which may be constructed as an inverse limit of $\mathbb{Z}/n\mathbb{Z}$.

Answer 2

The absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ is not even countable, see

Is the absolute Galois Group of $\Bbb Q$ countable?