{
  "id": "1704.03451",
  "version": "v1",
  "published": "2017-04-11T17:57:46.000Z",
  "updated": "2017-04-11T17:57:46.000Z",
  "title": "The least unramified prime which does not split completely",
  "authors": [
    "Asif Zaman"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K/F$ be a finite extension of number fields of degree $n \\geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\\mathbb{Q}$ and does not ramify or split completely in $K$. We improve upon the previous best known general estimates due to X. Li when $F = \\mathbb{Q}$ and Murty-Patankar when $K/F$ is Galois. Our bounds are the first when $K/F$ is not assumed to be Galois and $F \\neq \\mathbb{Q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-11T17:57:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R44"
    ],
    "keywords": [
      "unramified prime",
      "effective field-uniform unconditional upper bounds",
      "establish effective field-uniform unconditional upper",
      "prime ideal",
      "number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}