Shapes of pure prime degree number fields

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.