Scholarlink

<p>In this article, some generalizations of Girstmair’s irreducibility criterion have been established for polynomials having integer coefficients. These results for the ring of polynomials over the integers accentuate significant bounds on the number of irreducible factors of the underlying polynomial <i>f</i> apart from irreducibility under some austere factorization and divisibility conditions imposed on the integers <i>f</i>(<i>m</i>) and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2184_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(f^{(i)}(m)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the <i>i</i>-th formal derivative of <i>f</i> at <i>m</i> strictly superceding the height <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2184_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_f\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> of <i>f</i>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Further generalizations of Girstmair’s irreducibility criterion

  • 期刊论文

  • 发表日期：

    2025年10月15日

查看全文
  • Rishu Garg,
  • Jitender Singh,
  • Sanjeev Kumar

摘要

In this article, some generalizations of Girstmair’s irreducibility criterion have been established for polynomials having integer coefficients. These results for the ring of polynomials over the integers accentuate significant bounds on the number of irreducible factors of the underlying polynomial f apart from irreducibility under some austere factorization and divisibility conditions imposed on the integers f(m) and \(f^{(i)}(m)\) f ( i ) ( m ) , the i-th formal derivative of f at m strictly superceding the height \(H_f\) H f of f.