{
  "abstract": "Let $L/K$ be a degree $p$ extension of $p$-adic fields. We shall provide a complete characterization for the freeness of any fractional ideal $\\mathfrak{p}_L^h$, $h\\in\\mathbb{Z}$, of $L$ as a module over its associated order $\\mathfrak{A}_{L/K}(h)$ in the only Hopf-Galois structure on $L/K$, and an explicit description of $\\mathfrak{A}_{L/K}(h)$. We show that there is no loss of generality in assuming that $0\\leq h\\leq p-1$ and that the normal closure of $L/K$ is totally ramified over $K$. If $L/K$ is maximally ramified, then $\\mathfrak{A}_{L/K}(h)$ is the maximal $\\mathcal{O}_K$-order in $H$ and $\\mathfrak{p}_L^h$ is $\\mathfrak{A}_{L/K}(h)$-free. The remaining cases correspond to typical extensions. For these, we provide an $\\mathcal{O}_K$-basis of $\\mathfrak{A}_{L/K}(h)$ and a characterization for the condition that $\\mathfrak{p}_L^h$ is $\\mathfrak{A}_{L/K}(h)$-free. In the non-exceptional cases, which had been partially studied in the Galois setting, the characterization depends on $h$ and the continued fraction expansion of $\\frac{a}{p}$, where $a$ is the residue class mod $p$ of the (fractional) ramification jump $\\ell$ of $L/K$. In the remaining exceptional case, where $L/K$ is almost maximally ramified and $a+1\\leq h\\leq p-1$, it consists in a numerical common minimizer condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02301",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel Gil-Muñoz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1twgbfrd470n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02301",
  "fulltext_url": null,
  "id": "b1twgbfrd470n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02301",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1twgbfrd470n.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The module structure of ideals in degree $p$ extensions of $p$-adic fields",
  "title_text": "The module structure of ideals in degree p extensions of p-adic fields",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/b1twgbfrd470n",
  "venue": "arXiv",
  "versions": []
}