= Ramified prime classes for discriminant minus four d
{title2=$C_2^{\,k-1}\le\operatorname{Cl}(\mathbb Q(\sqrt{-d}))$}
Let $d>1$ be a positive <square-free integer> congruent to one modulo four, with distinct prime factors $p_1,\ldots,p_k$. The <ring of integers> is $\mathbb Z[\sqrt{-d}]$. Each ramified <prime ideal> $P_i=(p_i,\sqrt{-d})$ satisfies $P_i^2=(p_i)$, so its <ideal class> has order dividing two. Their product is the principal ideal $(\sqrt{-d})$.
There are no further subset relations. For a nonempty proper subset $S$, put $m=\prod_{i\in S}p_i$. If $\prod_{i\in S}P_i=(u+v\sqrt{-d})$, the <ideal norm> gives $m=u^2+dv^2$. But $1<m<d$ forces $v=0$ and makes the square-free integer $m$ a square, a contradiction. Thus the homomorphism $\mathbb F_2^k\to\operatorname{Cl}(K)$ defined by the ramified classes has exactly the all-ones relation, and its image is $C_2^{k-1}$. This identifies a subgroup, without asserting that these classes exhaust the <ideal class group>.
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