In a book, I find the following thing :
The natural homomorphism $ \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \to \displaystyle \lim_ {\longleftarrow n} \mathrm{Gal} (K_n / \mathbb{Q}) $ is an isomorphism, where $ K_n $ is a increasing sequence of Galois extensions of $ \mathbb {Q} $ such that the union of $ K_n $ is $ \overline{\mathbb{Q}} $.
Could you explain why? How is this isomorphism defined ?
Thank you in advance.
An element $\sigma$ of $ \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q})$ is an automorphism of $\overline{\mathbb{Q}}$ that fixes $\mathbb{Q}$.
First, note that restricting $\sigma$ to a field $K_n$ yields an automorphism of $K_n$ that fixes $\mathbb{Q}$ (recall here that since $K_n$ is Galois we have $\sigma(K_n) \subset K_n $), an element of $\mathrm{Gal} ( K_n / \mathbb{Q})$.
Hence, how to get a homomorphism $ \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \to \displaystyle \lim_ {\longleftarrow n} \mathrm{Gal} (K_n / \mathbb{Q}) $ is clear, just restrict the $\sigma$ for each $K_n$.
To see that it is injective, observe that if $\sigma_1 \neq \sigma_2$, then there is some $x \in \overline{\mathbb{Q}} $ such that $\sigma_1(x) \neq \sigma_2 (x)$. This $x$ is contained in some $K_n$ and so the restriction of $\sigma_1, \sigma_2$ to $K_n$ are different.
To see it is surjective, observe that for an element of the inverse limit you can define an automorphism $\sigma$ of $\overline{\mathbb{Q}}$ that fixes $\mathbb{Q}$ by saying $\sigma(x)= \sigma_m(x)$ where $m$ is such that $x \in K_m$. There is some choice here as $x$ is in more than one $K_m$, and one might worry there could be some incompatibility, yet the inverse limit is precisely defined just in such a way that there is no incompatibility.