# Hilbert class field whose class number is 1.

How to describe Hilbert class field of an imaginary quadratic field whose class number is 1 ? What happens to unramification at finite places ?

In general, if $K$ has class number one, then the ring of integers $\mathcal{O}_K$ is a PID, hence an UFD. In this case, $K$ is its own Hilbert class field - see here. Indeed, a number field $K$ with class number $1$ cannot have an abelian unramified extension larger than $K$ itself.