# is galois group always finite?

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

• Indeed $|Aut_{\mathbb{Q}} \mathbb{R}|=1$, but $\mathbb{R}$ is not a Galois extension of $\mathbb{Q}$ because, for example, it contains only one cube root of $2$. As long as $\mathbb{F}$ is a Galois extension of $\mathbb{Q}$, then you can use the formula in the answer of @ThomasGrubb, that $|Aut_{\mathbb{Q}} \mathbb{F}|$ equals the degree of $\mathbb{F}$ over $\mathbb{Q}$. Nov 16, 2017 at 17:47

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 $\mathbb{Z}/n\mathbb{Z}$.
• what is the notation $\Bbb F_p$ means ? Nov 16, 2017 at 17:27
The absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ is not even countable, see
Is the absolute Galois Group of $\Bbb Q$ countable?
• @user479583 that is not correct; $\overline{Q}$ is the set of all numbers which satisfy a polynomial with $\overline{Q}$ coefficients. For instance, $\sqrt{2}$ satisfies $x^2-2$. However, there are real numbers which satisfy no polynomials; we call them transcendental. For instance, $\pi$ and $e$ are transcendental. And in fact, there are algebraic numbers which are not even real! Perhaps you have heard of the complex number $i$, which satisfies $i^2=-1$. Nov 16, 2017 at 17:50
• thanks @ThomasGrubb , now I understand $\overline {\Bbb Q}$ means the algebraic closure of $\Bbb Q$ . not the closure we call in topology Nov 17, 2017 at 4:35
• @user479583 Equallly $\overline{\mathbb{F}_p}$ is the algebraic closure in in the answer of Thomas, appearing in the absolute Galois group of a finite field. Nov 17, 2017 at 10:05