关于带非平方自由参数的纯八次数域的整基与单本性

纯数域的单生成性：在典范数系存在性中的应用

On the monogenity of pure number fields: application to the existence of canonical number systems

本文在不假设 $m$ 无平方因子的条件下，利用 Ore 定理研究纯数域 $\mathbb{Q}(\sqrt[n]{m})$ 的单生成性，并应用于典范数系的存在性，推广了已有结果。

Integral bases and monogenity of composite fields

We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, o…

利用椭圆曲线确定纯三次数域的单源性

Determining monogenity of pure cubic number fields using elliptic curves

该研究利用椭圆曲线的 3-同源 Selmer 群刻画纯三次数域单源性，提出排除判据并给出单源域数量界，且在 GRH 下判定多个具体域。

On monogenity of certain pure number fields defined by $x^{2^u\cdot 3^v\cdot 5^t}-m$

Let $K = \mathbb{Q} (α) $ be a pure number field generated by a root $α$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v\cdot 5^t}-m$, with $ m \neq \pm 1 $ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md4$, $m\not\equiv \pm 1\md9$, and $m\not\in\{\pm 1, \pm 7\}\md{25}$, then $K$ is monogenic. But if {$m\equiv 1\md{4}$} or $m\equiv 1\md9$ or $m\equiv -1\md9$ and $u=2k$ for some odd integer $k$ or $u\ge 2$ and $m\equiv 1\md{25}$ or $m\equiv -1\md{25}$ and $u=2k$ for some odd in…

On power integral bases of certain pure number fields defined by $x^{2^u\cdot3^v}-m$

Let $K = \mathbb{Q} (α) $ be a pure number field generated by a complex root $α$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v}-m$, with $m \neq \pm 1$ a square free rational integer, $u$, and $v$ two positive integers. In this paper, we study the monogenity of $K$. The case $u=0$ or $v=0$ has been previously studied. We prove that if $m\not\equiv 1$ (mod4) and $m\not\equiv \pm 1$ (mod9), then $K$ is monogenic. But if $m\equiv 1$ (mod4) or $m\equiv 1$ (mod9), then $K$ is not monogenic. Some illustrating examples are given too.