Shapes of pure prime degree number fields

dc.contributor.advisor Harron, Robert
dc.contributor.advisor Varma, Ila Holmes, Erik
dc.contributor.department Mathematics 2021-09-30T18:15:19Z 2021-09-30T18:15:19Z 2021 Ph.D.
dc.subject Mathematics
dc.subject Algebraic number theory
dc.subject Arithmetic Statistics
dc.title Shapes of pure prime degree number fields
dc.type Thesis
dcterms.abstract This thesis extends a result of Rob Harron, in \cite{purecubics}. Specifically, Harron studies the shapes of pure cubic number fields $K=\QQ(\sqrt[3]{m})$ and shows that the shape is a complete invariant of the family of pure cubic number fields, and that the shapes are equidistributed on one-dimensional subspaces of the space of shapes. For pure prime degree number fields, $K=\QQ(\sqrt[p]{m})$, we show that the shape is a complete invariant. For $\ell=\frac{p-1}{2}$, our main result shows that the shapes of these fields lie on one of two $\ell$-dimensional subspaces of the space of shapes and we prove equidistribution results for $p<1000$. This work uses analytic methods which differ from those used in \cite{purecubics} and we therefore obtain an alternative proof of his result as well. We also prove that the family of pure prime degree number fields is equivalently the family of degree $p$ number field with Galois group $F_p$ and fixed resolvent field $\QQ(\zeta_p)$. This allows us to rephrase our results in a manner more closely related to those in the study of number field asymptotics and specifically Malle's conjecture. This alternative also allows us to ask a very natural follow up question which we intend to investigate in future work.
dcterms.extent 84 pages
dcterms.language en
dcterms.publisher University of Hawai'i at Manoa
dcterms.rights All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner.
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