Scholarlink

<p>Rational transformations play an important role in the construction of irreducible polynomials over finite fields. Usually, the methods involve fixing a rational function <i>Q</i> and deriving conditions on polynomials <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1591_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(F\in \mathbb {F}_q[x]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that the rational transformation of <i>F</i> with <i>Q</i> is irreducible. Here we want to change the perspective and study rational functions with which the rational transformation never yields irreducible polynomials. We show that if the rational function is contained in certain subfields of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1591_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {F}_q(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> then the rational transformation with it is always reducible. This extends the list of known examples.</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

Rational transformations over finite fields that are never irreducible

  • 期刊论文

  • 发表日期：

    2025年4月24日

查看全文
  • Max Schulz

摘要

Rational transformations play an important role in the construction of irreducible polynomials over finite fields. Usually, the methods involve fixing a rational function Q and deriving conditions on polynomials \(F\in \mathbb {F}_q[x]\) F ∈ F q [ x ] such that the rational transformation of F with Q is irreducible. Here we want to change the perspective and study rational functions with which the rational transformation never yields irreducible polynomials. We show that if the rational function is contained in certain subfields of \(\mathbb {F}_q(x)\) F q ( x ) then the rational transformation with it is always reducible. This extends the list of known examples.