{
  "id": "1810.11642",
  "version": "v1",
  "published": "2018-10-27T13:40:22.000Z",
  "updated": "2018-10-27T13:40:22.000Z",
  "title": "Some permutations over ${\\mathbb F}_p$ concerning primitive roots",
  "authors": [
    "Li-Yuan Wang",
    "Hao Pan"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime and let ${\\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\\mathbb F}_p$. Define the permutation $\\tau_g:\\,{\\mathcal H}_p\\to{\\mathcal H}_p$ by $$ \\tau_g(b):=\\begin{cases} g^b,&\\text{if }g^b\\in{\\mathcal H}_p,\\\\ -g^b,&\\text{if }g^b\\not\\in{\\mathcal H}_p,\\\\ \\end{cases} $$ for each $b\\in{\\mathcal H}_p$, where ${\\mathcal H}_p=\\{1,2,\\ldots,(p-1)/2\\}$ is viewed as a subset of ${\\mathbb F}_p$. In this paper, we investigate the sign of $\\tau_g$. For example, if $p\\equiv 5\\pmod{8}$, then $$ (-1)^{|\\tau_g|}=(-1)^{\\frac{1}{4}(h(-4p)+2)} $$ for every primitive root $g$, where $h(-4p)$ is the class number of the imaginary quadratic field ${\\mathbb Q}(\\sqrt{-4p})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-27T13:40:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concerning primitive roots",
      "permutation",
      "imaginary quadratic field",
      "class number",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}