{
  "abstract": "The index of a given monic irreducible polynomial $f(x)\\in\\mathbb Z[x]$ having a root $θ$ is the index of $\\mathbb Z[θ]$ in the ring of algebraic integers of $\\mathbb Q(θ).$ Further, $f(x)$ is monogenic if its index is $1$. In this article, we prove that the index of $f(x)$ always divides the index of $f(g(x)),$ where $f(x), g(x)\\in \\mathbb Z[x]$. Additionally, we give the precise power of the index of $f(x)$ dividing the index of $f(g(x))$. Also, we provide a necessary condition for the monogenity of $f(g(x))$. As an application of our results, we prove that the $n$-fold composition $f^n(x)=(f\\circ f\\cdots\\circ f)(x)$ of a polynomial $f(x)$ is monogenic for all $n\\in\\mathbb{N}$ if and only if the sequence $\\langle I_n\\rangle$ is convergent, where $I_n$ is the index of $f^n(x)$. Moreover, we show that for a given $n\\in\\mathbb{N}$, if $I_n$ is $m^{\\text{th}}$ power free integer, then the $n$-tower $\\mathbb Q\\subseteq K_1 \\subseteq\\cdots \\subseteq K_{n-1} \\subseteq K_n,$ defined by $n$ iterates of a polynomial $f(x)\\in\\mathbb Z[x]$ contains at least $\\max\\{\\lfloor n-\\log_{°f}m \\rfloor,0\\}$ monogenic number fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36877",
  "author_total": 2,
  "authors": [
    {
      "name": "Surender Kumar"
    },
    {
      "name": "Sumandeep Kaur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/csenqaeg9y0n7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36877",
  "fulltext_url": null,
  "id": "csenqaeg9y0n7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36877",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214986806",
  "pdf_url": "https://arxiv.org/pdf/2609.36877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/csenqaeg9y0n7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Index of Composition of polynomials and its applications",
  "title_text": "The Index of Composition of polynomials and its applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/csenqaeg9y0n7",
  "venue": "arXiv",
  "versions": []
}