I am trying to understand infinite galois theory by self-made examples. First example I am struggling with is the field of all constructible (by compass and straightedge) complex numbers - let's call it $\mathbb{M}$. Well known fact is, that every such constructible number has degree $2^k$ over $\mathbb{Q}$ for some $k\in \mathbb{N}$. In fact there is, for every $\alpha\in\mathbb{M}$ a chain of fields $\mathbb{Q} = K_0 \subset K_1 \subset \dots K_m$ with $\alpha\in K_m$ and $[K_n:K_{n-1}] = 2\ \forall n=1, \dots, m$.
I do wonder now, what the Galois group of $\mathbb{M}/\mathbb{Q}$ looks like. I am aware of the construction of the Krull topology on the projective limit of all finite Galois subextensions, but precisely in order to understand this construction, I am making this example.
So here's my first question, from which I hope I can proceed to understand: can someone name a simple example of an element of the Galois group? Is for example a map, that maps each instance of $\sqrt2$ to $-\sqrt2$ a $\mathbb{Q}$-automorphism of $\mathbb{M}$ and therefore an element of the group? If so, how to show that? Part of the problem here seems to be, that I don't even know, how to write the elements of $\mathbb{M}$.
Now if this is the case and my map is an element of $Gal(\mathbb{M}/\mathbb{Q})$, how does it act on intermediate galois fields like $\mathbb{Q}(\sqrt6)$ or $\mathbb{Q}(i, \sqrt{\sqrt{2}+1})$? Are these expressions "opaque" for this automorphism or is it rather as I (naively) expected, that each number that "contains" $\sqrt2$ is affected by this element?
What I want to get at is this: suppose we fix some intermediate, finite Galois extension fields as the first components of the infinite Cartesian product of which the projective limit is a subset, say $\mathbb{Q}(\sqrt2)$ as the first, then $\mathbb{Q}(\sqrt3)$, $\mathbb{Q}(\sqrt6)$ and $\mathbb{Q}(i, \sqrt{\sqrt{2}+1})$. The isomorphism between this product and $Gal(\mathbb{M}/\mathbb{Q})$ maps an element $\varphi \in Gal(\mathbb{M}/\mathbb{Q})$ to the vector of the restrictions of $\varphi$ to all finite Galois extension fields. So now, what is the image of a possible $\varphi: (\sqrt2 \mapsto -\sqrt2)$ under this isomorphism in the first components?
I am hoping that in the long run, this example will enable me, to understand, which fields are open and which closed in the Krull topology.