The folklore conjecture on unramified extensions of quadratic number fields

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The folklore conjecture on unramified extensions of quadratic number fields

   Let $G$ be a finite group, and let $K$ range over quadratic number fields. An extension of $K$ is unramified at all primes when it is unramified at every finite and infinite prime of $K$. Folklore conjecture. For any finite group $G$, there exist infinitely many quadratic number fields $K$ such that $K$ possesses a Galois extension with Galois group $G$ unramified at all finite and infinite primes of $K$. This conjecture concerns the existence of everywhere-unramified extensions with prescribed finite Galois group; the paper establishes such infinite families for the group $SL_2(5)$, but the general assertion remains open.

   source: Joachim König, “Quadratic number fields with unramified SL_2(5)-extensions”, arXiv:2211.01555 (2022).