论文标题
在具有非平方参数的纯八核数字段的整体基础和单基因上
On integral bases and monogenity of pure octic number fields with non-square free parameters
论文作者
论文摘要
在所有可用的论文中，在纯八粒字段$ k $的电源积分基础上，由root $α$产生的monic不可验证的多项式$ f（x）= x^8-m \ in \ mathbf z [x] $，假定$ m \ m \ neq \ pm pm 1 $是免费的。在本文中，我们研究了任何纯八核数字字段的单基因，而没有$ m $无方形的条件。我们首先计算$ k $ $ k $的整数环$ \ mathbf z_k $的积分基础。特别是，我们表征$ \ mathbf z_k = \ mathbf z [α] $。我们对$ m $提供足够的条件，这可以保证$ k $不是单基因的。当$ m = a^u $，$ u \ in \ {1,3,5,7 \} $和$ a \ neq \ mp 1 $时，我们通过调查案件来完成论文。
In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $α$ of a monic irreducible polynomial $f(x)=x^8-m\in\mathbf Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We start by calculating an integral basis of $\mathbf Z_K$, the ring of integers of $K$. In particular, we characterize when $\mathbf Z_K=\mathbf Z[α]$. We give sufficient conditions on $m$, which guarantee that $K$ is not monogenic. We finish the paper by investigating the case when $m=a^u$, $u\in\{1,3,5,7\}$ and $a\neq \mp 1$ is a square free rational integer.
