On the Index of Certain Nonic Number Fields Defined by $$x^9+ax^5+b$$

IF 0.5 Q4 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-05-06 DOI:10.1134/s2070046624020018

Omar Kchit

引用次数: 0

Abstract

In this paper, for any nonic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^9+ax^5+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz [23] for this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.

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论由 $$x^9+ax^5+b$$ 定义的某些 Nonic 数域的索引

摘要在本文中，对于由一元不可还原三项式$F(x)=x^9+ax^5+b \in \mathbb{Z}[x]$ 的根$\alpha$产生的任何数域$K$，以及对于每个有理素数$p$，我们描述了当$p$除以$K$的索引时的特征。我们还描述了索引（i(K)\）的素幂分解。通过这种方式，我们给出了纳基维茨（Narkiewicz）[23]针对这个数域族提出的问题（22）的部分答案。作为我们结果的应用，如果 $i(K)\neq1$, 那么 $K$ 不是单源的。我们通过一些计算实例来说明我们的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

P-Adic Numbers Ultrametric Analysis and Applications MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-

CiteScore

1.10

自引率

20.00%

发文量

期刊介绍： This is a new international interdisciplinary journal which contains original articles, short communications, and reviews on progress in various areas of pure and applied mathematics related with p-adic, adelic and ultrametric methods, including: mathematical physics, quantum theory, string theory, cosmology, nanoscience, life sciences; mathematical analysis, number theory, algebraic geometry, non-Archimedean and non-commutative geometry, theory of finite fields and rings, representation theory, functional analysis and graph theory; classical and quantum information, computer science, cryptography, image analysis, cognitive models, neural networks and bioinformatics; complex systems, dynamical systems, stochastic processes, hierarchy structures, modeling, control theory, economics and sociology; mesoscopic and nano systems, disordered and chaotic systems, spin glasses, macromolecules, molecular dynamics, biopolymers, genomics and biology; and other related fields.