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arXiv · 2209.10638

On the shapes of pure prime degree number fields

Erik Holmes
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Abstract

For $p$ prime and $\ell = \frac{p-1}{2}$, we show that the shapes of pure prime degree number fields lie on one of two $\ell$-dimensional subspaces of the space of shapes, and which of the two subspaces is dictated by whether or not $p$ ramifies wildly. When the fields are ordered by absolute discriminant we show that the shapes are equidistributed, in a regularized sense, on these subspaces. We also show that the shape is a complete invariant within the family of pure prime degree fields. This extends the results of Harron, in [Har17], who studied shapes in the case of pure cubic number fields. Furthermore we translate the statements of pure prime degree number fields to statements about Frobenius number fields, $F_p = C_p\rtimes C_{p-1}$, with a fixed resolvent field. Specifically we show that this study is equivalent to the study of $F_p$-number fields with fixed resolvent field $\mathbb{Q}(\zeta_p)$.

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Erik Holmes. 2022-09-21. On the shapes of pure prime degree number fields. https://arxiv.org/abs/2209.10638

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