It is well known that the absolute Galois group of $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{q})$ are nonisomorphic if $p$ and $q$ are different prime numbers.
See for example Szamuely's book "Galois groups and fundamental groups".
My question is, how do those Galois groups $\mathrm{Gal}(\overline{\mathbb{Q}(\sqrt{p})}/\mathbb{Q}(\sqrt{p}))$ look like?
Edit. The question seems to be much more complicated that I expected, in fact problably an open problem. I include a weakened version of the question.
Can we say anything non-trivial and specific about $\mathrm{Gal}(\overline{\mathbb{Q}(\sqrt{2})}/\mathbb{Q}(\sqrt{2}))$ as a profinite group?
For the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ it is very difficult to say something non-trivially as profinite group, see for example the notes of Leila Schneps. One would like to find characterizations of the elements of this group via its geometric actions. This is not possible directly. On page $2$ she says "Apart from complex conjugation, it is impossible to ‘write down’ an element of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$". I am not sure that the situation becomes much easier for the Galois groups of number fields, or global fields in general.