Principal factors and lattice minima

Abstract

Let $k=\mathbb{Q}(\sqrt[3]{d},{\zeta }_3)$, where $d>1$ is a cube-free positive integer, $k_0=\mathbb{Q}({\zeta }_3)$ be the cyclotomic field containing a primitive cube root of unity ${\zeta }_3$, and $G=\mathrm{Gal}(k/k_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $\Theta =({\theta }_i{)}_{i\in \mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{d})$ of $k$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(\alpha )\in {\mathcal{P}}_k^G/{\mathcal{P}}_{k_0}$ among the lattice minima $\Theta =({\theta }_i{)}_{i\in \mathbb{Z}}$ of the underlying pure cubic field $L=\mathbb{Q}(\sqrt[3]{d})$, and to explain exceptional behavior of the chain $\Theta$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $k$ by means of Voronoi's algorithm.

17 pages, 5 Tables