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

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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.

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