The shapes of pure cubic fields

TL;DR

This paper classifies the geometric shapes of pure cubic fields, showing they form two families with distinct shape properties, and demonstrates that these shapes uniquely identify the fields within all cubic fields.

Contribution

It provides a complete classification of the shapes of pure cubic fields and proves that these shapes uniquely determine the fields among all cubic fields.

Findings

Shapes of Type I fields are rectangular and equidistributed.

Shapes of Type II fields are non-rectangular but follow similar distribution.

Shape acts as a complete invariant for pure cubic fields.

Abstract

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are rectangular and that they are equidistributed, in a regularized sense, when ordered by discriminant, in the one-dimensional space of all rectangular lattices. We do the same for Type II fields, which are however no longer rectangular. We obtain as a corollary of the determination of these shapes that the shape of a pure cubic field is a complete invariant determining the field within the family of all cubic fields.