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The shapes of pure cubic fields


Author: Robert Harron
Journal: Proc. Amer. Math. Soc. 145 (2017), 509-524
MSC (2010): Primary 11R16, 11R45, 11E12
DOI: https://doi.org/10.1090/proc/13309
Published electronically: August 18, 2016
MathSciNet review: 3577857
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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.


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Additional Information

Robert Harron
Affiliation: Department of Mathematics, Keller Hall, University of Hawai‘i at Mānoa, Honolulu, Hawaii 96822
Email: rharron@math.hawaii.edu

DOI: https://doi.org/10.1090/proc/13309
Received by editor(s): September 4, 2015
Received by editor(s) in revised form: April 10, 2016
Published electronically: August 18, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society