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

IF 0.5 Q4 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Omar Kchit
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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.

Abstract Image

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。
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来源期刊
P-Adic Numbers Ultrametric Analysis and Applications
P-Adic Numbers Ultrametric Analysis and Applications MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-
CiteScore
1.10
自引率
20.00%
发文量
16
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.
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