Is there a general formula for the number of unramified quadratic extensions of a number field $K$?

When $K$ is quadratic, this is known (by genus theory) to be $2^{\omega(\Delta_K)-1}$, where $\omega(n)$ denotes the number of distinct prime factors of $n$ and $\Delta_K$ is the discriminant of $K$. I'm interested in results for when $K$ is of higher degree.

It seems like this problem might be much harder and is maybe adjacent to understanding the two-torsion of the class group $\text{Cl}_K$ (which seems hard when $K$ is not quadratic), but I'm pretty new to the area and could be totally off-base. Is there any hope of a more direct approach?

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