{
  "id": "2307.04912",
  "version": "v1",
  "published": "2023-07-10T21:34:51.000Z",
  "updated": "2023-07-10T21:34:51.000Z",
  "title": "A Generalization of Arithmetic Derivative to $p$-adic Fields and Number Fields",
  "authors": [
    "Brad Emmons",
    "Xiao Xiao"
  ],
  "comment": "Submitted",
  "categories": [
    "math.NT"
  ],
  "abstract": "The arithmetic derivative is a function from the natural numbers to itself that sends all prime numbers to $1$ and satisfies the Leibniz rule. The arithmetic partial derivative with respect to a prime $p$ is the $p$-th component of the arithmetic derivative. In this paper, we generalize the arithmetic partial derivative to $p$-adic fields (the local case) and the arithmetic derivative to number fields (the global case). We study the dynamical system of the $p$-adic valuation of the iterations of the arithmetic partial derivatives. We also prove that for every integer $n\\geq 0$, there are infinitely many elements with exactly $n$ anti-partial derivatives. In the end, we study the $p$-adic continuity of arithmetic derivatives.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-10T21:34:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11R04"
    ],
    "keywords": [
      "arithmetic derivative",
      "adic fields",
      "number fields",
      "arithmetic partial derivative",
      "generalization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}