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Abstract <p> Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(K\)</EquationSource> </InlineEquation> be a number field with ring of algebraic integers <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(R\)</EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L=K(\alpha)\)</EquationSource> </InlineEquation> be a finite extension of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(K\)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> is a root of an irreducible polynomial of the type <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f(x)=x^n+ax^m-b\)</EquationSource> </InlineEquation> belonging to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(R[x]\)</EquationSource> </InlineEquation> (<InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(n, m\in \mathbb{N}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(n&gt;m \)</EquationSource> </InlineEquation>). In this paper, we give a set of necessary and sufficient conditions to study the monogenity of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(L\)</EquationSource> </InlineEquation> over <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(K\)</EquationSource> </InlineEquation>. </p> <p> We also provide a class of finite separable extensions <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(L\)</EquationSource> </InlineEquation> of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(K\)</EquationSource> </InlineEquation> for which <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(L/K\)</EquationSource> </InlineEquation> is not monogenic. Our results extend the one given in [<CitationRef CitationID="CR1">1</CitationRef>]. </p>

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A Note on Relative Power Integral Basis of Number Fields Defined by \(x^n + ax^m - b\)

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • S. Kaur,
  • A. Jakhar

摘要

Abstract

Let \(K\) be a number field with ring of algebraic integers \(R\) . Let \(L=K(\alpha)\) be a finite extension of \(K\) where \(\alpha\) is a root of an irreducible polynomial of the type \(f(x)=x^n+ax^m-b\) belonging to \(R[x]\) ( \(n, m\in \mathbb{N}\) and \(n>m \) ). In this paper, we give a set of necessary and sufficient conditions to study the monogenity of \(L\) over \(K\) .

We also provide a class of finite separable extensions \(L\) of \(K\) for which \(L/K\) is not monogenic. Our results extend the one given in [1].