Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact?
Suppose we have an number field $K$, is any Galois extension of $K$ ramified? I think the answer is no, but I do not have a clear picture, examples will be appreciated.
My main question is the following:
Can we reconstruct the absolute Galois group $\mathrm{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ using only unramified Galois extensions?
Best Answer
I am far from being an expert, but I can confirm that there exist number fields $K\neq\mathbb{Q}$ which have no nontrivial unramified extensions. For example, imaginary quadratic number fields of class number one have this property. See this paper by Yamamura for more examples and background information.
On the other hand, I don't understand your main question. What do you mean by "reconstructing" the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$?