Search ResultsShowing 1-3 of 3
-
arXiv:2505.06213 (Published 2025-05-09)
Determining monogenity of pure cubic number fields using elliptic curves
Categories: math.NTWe study monogenity of pure cubic number fields by means of Selmer groups of certain elliptic curves. A cubic number field with discriminant $D$ determines a unique nontrivial $\mathbb{F}_3$-orbit in the first cohomology group of the elliptic curve $E^D: y^2 = 4x^3 + D$ with respect to a certain 3-isogeny $\phi$. Orbits corresponding to monogenic fields must lie in the soluble part of the Selmer group $S^{\phi}(E^D/\mathbb{Q})$, and this gives a criterion to discard monogenity. From this, we can derive bounds on the number of monogenic cubic fields in terms of the rank of the elliptic curve. We can also determine the monogenity of many concrete pure cubic fields assuming GRH.
-
arXiv:2209.10638 (Published 2022-09-21)
On the shapes of pure prime degree number fields
Comments: 36 pages, 1 figure. Comments welcome!Categories: math.NTFor $p$ 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 $\mathbb{Q}(\zeta_p)$.
-
arXiv:1108.6069 (Published 2011-08-30)
Why is the Class Number of $\Q(\sqrt[3]{11})$ even?
Categories: math.NTIn this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.