In the first two articles of this series, we investigated various higher analogues

In the first two articles of this series, we investigated various higher analogues
of Gauss composition, and showed how several algebraic objects involving
orders in quadratic and cubic fields could be explicitly parametrized. In
particular, a central role in the theory was played by the parametrizations of
the quadratic and cubic rings themselves.
These parametrizations are beautiful and easy to state. In the quadratic
case, one need only note that a quadratic ring—i.e., any ring that is free of rank
2 as a Z-module—is uniquely specified up to isomorphism by its discriminant;
and conversely, given any discriminant D, i.e., any integer congruent to 0 or 1
(mod 4), there is a unique quadratic ring having discriminant D, namely
S(D) =


Z[x]/(x2) if D = 0,
Z · (1, 1) + √D(Z ⊕ Z) if D ≥ 1 is a square,
Z[(D + √D)/2] otherwise.