{
  "abstract": "In this paper, we investigate the existence of primitive polynomials over $\\mathbb{F}_{q^n}$ whose constant term is a primitive element of $\\mathbb{F}_{q^n}$. We prove that such polynomials do not exist if $q$ is odd, $q^n\\equiv3\\pmod{4}$, and the degree $m$ of the polynomial is odd. In particular, the polynomials $f(x)=g(x)+λ$, where $g(x)\\in\\mathbb{F}_q[x]$ satisfies $g(0)=0$ and $λ\\in\\mathbb{F}_{q^n}$ is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial $x^3+x^2+x+λ$, we establish non-existence results in characteristics $2$ and $3$. These results, in particular, provide counterexamples to previously proposed existence conjectures. We also study the family $x^p+x+λ$ over $\\mathbb{F}_{p^n}$. For an odd prime $p$, we prove that, provided $\\sum_{i=0}^{n-1}(-1)^iλ^{p^i}\\neq0$, the polynomial $x^p+x+λ$ is irreducible over $\\mathbb{F}_{p^n}$ if and only if $n$ is even. Motivated by this result and supported by computational evidence, we formulate a conjecture concerning the existence of such primitive polynomials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.07262",
  "author_total": 1,
  "authors": [
    {
      "name": "Avnish K. Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0q73jnzvbyfxq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.07262",
  "fulltext_url": null,
  "id": "0q73jnzvbyfxq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.07262",
  "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",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7202117008",
  "pdf_url": "https://arxiv.org/pdf/2608.07262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0q73jnzvbyfxq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.07262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.07262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Primitive Polynomials of the Form $g(x)+λ$ over Finite Fields: Non-Existence Results and Conjectures",
  "title_text": "Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0q73jnzvbyfxq",
  "venue": "arXiv",
  "versions": []
}