A generalization of a result of Barrucand and Cohn on class numbers (Q1310828)

The author generalizes a result of \textit{B. Barrucand} and \textit{H. Cohn} [J. Reine Angew. Math. 238, 67-70 (1969; Zbl 0207.362)] concerning the 8-rank of the class group of imaginary quadratic fields. Let \(L = L^{(1)} L^{(2)} \cdots L^{(v)}\) be a square free integer where each \(L^{(i)} \equiv 1 \pmod 8\) is a product of primes which are congruent to 1 modulo 4. Additionally, assume any two prime divisors of \(L^{(i)}\) are quadratic nonresidues of each other. Also, for each \(i\) with \(2 \leq i \leq v\), assume there is exactly one pair of primes \(p_ i\) and \(q_ i\) such that \(p_ i | L^{(i)} \cdots L^{(i - 1)}\) and \(q_ i | L^{(i)}\) which are quadratic nonresidues of each other. Let \(R\) denote the product of the prime divisors of \(L\) which are congruent to 5 modulo 8 and \(P\) denote the product of those primes which are congruent to 1 modulo 8. Moreover, write \(P = A^ 2 + B^ 2\) and \(R = C^ 2 + D^ 2\) with \(B\) and \(D\) even. The author proves that the ideal class group \(C_ K\) of the field \(K = \mathbb{Q} (\sqrt { - 4L})\) has 4-rank 1. Furthermore, \(C_ K\) has 8-rank 1 if and only if the Jacobi Symbols satisfy \((1 + \sqrt {-1}/P) (D - C/R)(AD + BC/R) = 1\).