Unramified Quadratic Extension of Number Field – Number Theory Question

Question

Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?

Answer 1

Ah... perhaps an amplification of my comment and @Aurel's would be useful to you:

First, this kind of thing is not really in Hilbert's "Zahlbericht", because he only treats a sub-class of extensions... which was already a novelty, etc.

But by the 1920's, Takagi and Artin had clarified/proved the reasonable general case of Hilbert's earlier examples, namely, that the Galois group of the maximal unramified abelian extension of a number field was naturally isomorphic to the absolute ideal class group of the ring of integers of that number field. This is a little bit more delicate than just the general assertions of classfield theory, since it requires further attention to ramification...

The general theorems of classfield theory are proven many places, but, as far as I know, the subtler versions, about ramification and "Hilbert classfields", are not reliably proven. (E.g., I think Lang's otherwise very useful "Alg No Th" does not actually prove that, though remarks upon it.)

Anyway, the facts are fairly straightforward! :)