Cyclic cubic monogenic extensions of algebraic integers of a quadratic field (Q1291143)

Let \(A_d\) denote the ring of integers of the quadratic number field \(\mathbb Q ( \sqrt d)\), \(d \neq 3\). This paper exhibits criteria for the existence of an irreducible, cubic polynomial \(P \in A_d [X]\) with cyclic Galois group and discriminant equal to \(1\). Any root of such a polynomial then generates the ring of integers of a relatively unramified, cyclic, cubic extension of \(\mathbb Q ( \sqrt d)\). The author checked that 3730 square-free integers \(d\) with \(|d|< 100 000\) satisfy these criteria, for 69 of them he found a polynomial \(P\) with the above properties.