{
  "id": "2203.07870",
  "version": "v1",
  "published": "2022-03-07T17:39:19.000Z",
  "updated": "2022-03-07T17:39:19.000Z",
  "title": "Fermat's Last Theorem over ${\\mathbb Q}(\\sqrt{5})$ and ${\\mathbb Q}(\\sqrt{17})$",
  "authors": [
    "Imin Chen",
    "Aisosa Efemwonkieke",
    "David Sun"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove Fermat's Last Theorem over ${\\mathbb Q}(\\sqrt{5})$ and ${\\mathbb Q}(\\sqrt{17})$ for prime exponents $p \\ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\\mathbb Q}(\\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\\mathbb Q}(\\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-07T17:39:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41"
    ],
    "keywords": [
      "real quadratic base field",
      "congruence classes modulo",
      "real quadratic fields",
      "quadratic reciprocity constraints",
      "modular method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}