On integral bases and monogenity of pure octic number fields with non-square free parameters

Abstract

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.

Figures (1)

• Figure 1: $N_{\phi}^-(f)$.

Theorems & Definitions (14)

• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5