Monogenic cubic cyclic extension of the ring of integers of a quadratic field
Let \(A\) be the ring of integers of the quadratic field \(\mathbb{Q} (\sqrt d)\), \(d\neq -3\). In [Arch. Math. 61, No. 4, 348-361 (1993; Zbl 0789.11058)] the author showed that if \(A\) has a cubic, cyclic, monogenic extension of discriminant one, then either of two diophantine equations must have a solution in \(A\). In this paper the author shows that only one of the equations has solutions which yields such extensions: \(A\) has such an extension iff \(4m^ 3= y^ 2 d+27\) has a solution in \(\mathbb{Z}\) with \(d\not\equiv 11, 21, 23, 35\pmod {36}\) or \(d\equiv 1\pmod {12}\) and \(m\not\equiv 3\pmod 9\). The defining polynomial and nontrivial automorphism of the extension are determined.
scientific article; zbMATH DE number 756421