The 2-Sylow Subgroup

One of the most carefully studied questions about quadratic class groups is the precise determination of the 2-Sylow subgroup. This is intimately connected, in the case of positive discriminant, with the question of which discriminants Δ possess solutions of the negative Pell equation 9.1 $$ {X^2} - \Delta {Y^2} = - 4 $$ and both questions are related for all discriminants to the existence of higher-order reciprocity laws analogous to the law of quadratic reciprocity. The connections come from the following observations. Given a fundamental discriminant with t prime factors, we are guaranteed a factor of 2 t−1 in the class number. This is the exact power of 2 in the class number if and only if the 2-Sylow subgroup is an elementary 2-group, or if the genera each have an odd number of classes, or if there is exactly one ambiguous class per genus, these being equivalent conditions. The ambiguous classes correspond to factors of the discriminant, and a class which represents a prime p dividing Δ also represents Δ/p, so the 2-Sylow subgroup can only be elementary if no prime factor of Δ is a quadratic residue of all the other prime factors of Δ (For the sake of argument, we ignore for the moment the extra characters ϰ and ψ.)