Abstract
Let Δ( u) denote the discriminant of x 3 + ( u + 1) x 2 − ( 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]).
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