Class numbers of complex quadratic fields (Q1844258)

Using the theory of composition of binary quadratic forms, the author studies the class number \(h(-pq)\) of \(\mathbb Q(\sqrt{-pq})\), where \(p\) and \(q\) are distinct primes such that \(p\equiv q\pmod 4\). Among the results are the following: (a) if \(p\equiv q\equiv 1\pmod 4\), then \(h(-pq)\equiv 4\pmod 8\), if \(p\not\equiv q\pmod 8\) and \((p\mid q) = -1\); \(h(-pq) \equiv 0 \pmod{16}\) if (\((p\mid q) =1\), \(p\equiv q\equiv 1\pmod 8\); \(h(-qp) \equiv 0 \pmod 8\), otherwise. (b) If \(p\equiv q\equiv 7\pmod 8\) and \(pq=2e^2 - d^2\) with \(e> 0\), then \((p\mid q) =1\) implies \(h(-pq) \equiv 0\text{ or }8 \pmod{16}\) according as \((e\mid q) =1\) or \(-1\), and \((p\mid q) = (e\mid p) = (e\mid q) =1\) implies \(h(-pq) \equiv 0 \pmod{32}\).