8-ranks of class groups of some imaginary quadratic number fields

Let F = Q(root-p(1)p(2)) be an imaginary quadratic field with distinct primes p(1) = p(2) = 1 mod 8 and the Legendre symbol (p(1)/p(2)) = 1. Then the 8-rank of the class group of F is equal to 2 if and only if the following conditions hold: (1) The quartic residue symbols (p(1)/p(2))4 = (p(1)/p(2))4 = 1; (2) Either both p(1) and p(2) are represented by the form a(2) + 32b(2) over Z and p(2)(h+(2p1)/4) = x(2) - 2p(1)y(2), x, y is an element of Z, or both p(1) and p(2) are not represented by the form a(2) + 32b(2) over Z and p(2)(h+(2p1)/4) = epsilon(2x(2) - p(1)y(2)), x, y is an element of Z, epsilon is an element of {+/-1}, where h(+)(2p(1)) is the narrow class number of Q(root 2p(1)). Moreover, we also generalize these results.