= Class number divisibility for a CM extension
{title2=$h_{K^+}\mid h_K$}
For a <CM-field> $K$ over its maximal real subfield $K^+$, the ordinary <Hilbert class field> $H_{K^+}$ is totally real because its real places split. Thus $H_{K^+}\cap K=K^+$. Its <compositum> with $K$ is an abelian extension of $K$ of degree $h_{K^+}$, and is everywhere unramified by <unramified extensions remain unramified under base change>. It lies in $H_K$, giving the divisibility. The argument does not require injectivity of extension of <ideal classes>.
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