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Principal factors and lattice minima

Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},ζ_3)$, where $d>1$ is a cube-free positive integer, $\mathit{k}_0=\mathbb{Q}(ζ_3)$ be the cyclotomic field containing a primitive cube root of unity $ζ_3$, and $G=\operatorname{Gal}(\mathit{k}/\mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $Θ=(θ_i)_{i\in\mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{d})$ of $\mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(α)\in\mathcal{P}_{\mathit{k}}^G/\mathcal{P}_{\mathit{k}_0}$ among the lattice minima $Θ=(θ_i)_{i\in\mathbb{Z}}$ of the underlying pure cubic field $L=\mathbb{Q}(\sqrt[3]{d})$, and to explain exceptional behavior of the chain $Θ$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $\mathit{k}$ by means of Voronoi's algorithm.