= Class number divisibility with split real places
{title2=$h_A\mid h_B$}
If a finite extension $B/A$ of <number fields> has a totally ramified finite prime, the ordinary <Hilbert class field> $H_A$ is disjoint from $B$. Its base change $BH_A/B$ is abelian and unramified at finite primes. Its real places also split, since a split-real unramified extension retains this property under base change. It therefore lies in the ordinary <Hilbert class field> of $B$, proving $h_A\mid h_B$ even when $A$ has real places.
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