Many authors have
considered the divisibility of the restricted class number h+(d) of the quadratic field
Ω = Q() by 4 and 8, in the case that the discriminant d of Ω has exactly two
prime factors. For such discriminants the restricted classgroup 𝒞 of Ω has a
nontrivial cyclic 2-Sylow subgroup, and conditions on d can be given for the
existence of classes in 𝒞 of orders 4 and 8. The first such results are due to
Rédei.
In this paper we give criteria for the divisibility of h+(d) by 8 which are phrased
in terms of the splitting of one of the prime factors p of d in a normal extension of Q
depending only on d∕p = d0. This simplifies and unifies the criteria for 8∣h+(d)
existing in the literature, which depend mainly in the representation of the prime p
by certain quadratic forms, or on the quadratic character of solutions to ternary
quadratic equations.
