{
  "id": "2404.15063",
  "version": "v1",
  "published": "2024-04-23T14:11:21.000Z",
  "updated": "2024-04-23T14:11:21.000Z",
  "title": "On cyclotomic matrices involving Gauss sums over finite fields",
  "authors": [
    "Hai-Liang Wu",
    "Jie Li",
    "Li-Yuan Wang"
  ],
  "comment": "This is a preliminary version",
  "categories": [
    "math.NT"
  ],
  "abstract": "Inspired by the Carlitz's work on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields. For example, let $q=p^n$ be an odd prime power with $p$ prime and $n\\in\\mathbb{Z}^+$. Let $\\zeta_p=e^{2\\pi{\\bf i}/p}$ and let $\\chi$ be a generator of the group of all mutiplicative characters of the finite field $\\mathbb{F}_q$. For the Gauss sum $$G_q(\\chi^{r})=\\sum_{x\\in\\mathbb{F}_q}\\chi^{r}(x)\\zeta_p^{{\\rm Tr}_{\\mathbb{F}_q/\\mathbb{F}_p}(x)},$$ we prove that $$\\det \\left[G_q(\\chi^{2i+2j})\\right]_{0\\le i,j\\le (q-3)/2}=(-1)^{\\alpha_p}\\left(\\frac{q-1}{2}\\right)^{\\frac{q-1}{2}}2^{\\frac{p^{n-1}-1}{2}},$$ where $$\\alpha_p=\\begin{cases}1 & \\mbox{if}\\ n\\equiv 1\\pmod 2, (p^2+7)/8 & \\mbox{if}\\ n\\equiv 0\\pmod 2. \\end{cases}$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-23T14:11:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gauss sum",
      "finite field",
      "cyclotomic matrices",
      "odd prime power",
      "carlitzs work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}