On a Parameterized Family of Quadratic and Cubic Fields

AbstractLet Δ(u) denote the discriminant of x3 + (u + 1) x2 − (u + 2) x − 1. The polynomial Δ(u) factors in four distinct ways as a square minus four times a cube. We show that under certain congruence conditions on u the quadratic fields Q([formula]) always have class number divisible by 3. What we specifically prove is that the ideal splitting 2 in these fields is always of order 3 in the class group. We also prove that three other pairs of ideals of orders dividing 6 are principal only under certain congruence conditions module 14 and that for squarefree values of Δ(u) the 3-rank of the class group of Q([formula]) is the same as that of Q([formula])