{
  "id": "2407.20583",
  "version": "v1",
  "published": "2024-07-30T06:30:35.000Z",
  "updated": "2024-07-30T06:30:35.000Z",
  "title": "Jacobi sums and cyclotomic matrices involving squares over finite fields",
  "authors": [
    "Hai-Liang Wu",
    "Li-Yuan Wang"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $q=p^n$ be an odd prime power and let $\\mathbb{F}_q$ be the finite field of $q$ elements. Let $\\widehat{\\mathbb{F}_q^{\\times}}$ be the group of all multiplicative characters of $\\mathbb{F}_q$ and let $\\chi$ be a generator of $\\widehat{\\mathbb{F}_q^{\\times}}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over $\\mathbb{F}_q$. For example, let $s_1,s_2,\\cdots,s_{(q-1)/2}$ be the nonzero squares over $\\mathbb{F}_q$. For any integer $1\\le r\\le q-2$, define the matrix $$B_q(r):=\\left[\\chi^r(s_i+s_j)+\\chi^r(s_i-s_j)\\right]_{1\\le i,j\\le (q-1)/2}.$$ We prove that if $q\\equiv 3\\pmod 4$, then $$\\det (B_q(r))=\\prod_{0\\le k\\le (q-3)/2}J_q(\\chi^r,\\chi^{2k})= \\begin{cases} (-1)^{\\frac{q-3}{4}}{\\bf i}^nG_q(\\chi^r)^{\\frac{q-1}{2}}/\\sqrt{q} & \\mbox{if}\\ r\\equiv 1\\pmod 2, G_q(\\chi^r)^{\\frac{q-1}{2}}/q & \\mbox{if}\\ r\\equiv 0\\pmod 2. \\end{cases},$$ where $J_q(\\chi^r,\\chi^{2k})$ and $G_q(\\chi^r)$ are the Jacobi sum and the Gauss sum over $\\mathbb{F}_q$ respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-30T06:30:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "jacobi sum",
      "cyclotomic matrices",
      "finite field",
      "nonzero squares",
      "odd prime power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}