{
  "id": "0911.5472",
  "version": "v3",
  "published": "2009-11-29T09:41:05.000Z",
  "updated": "2010-09-28T08:34:54.000Z",
  "title": "Complete Solving for Explicit Evaluation of Gauss Sums in the Index 2 Case",
  "authors": [
    "Jing Yang",
    "Lingli Xia"
  ],
  "journal": "SCIENCE CHINA Mathematics, 2010, Sep., 53(9), 2525--2542",
  "doi": "10.1007/s11425-010-3155-z",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\\gcd(N,p)=1$, and let $\\k$ be a primitive multiplicative character of order $N$ over finite field $\\fq$. This paper studies the problem of explicit evaluation of Gauss sums in \"\\textsl{index 2 case}\" (i.e. $f=\\f{\\p(N)}{2}=[\\zn:\\pp]$, where $\\p(\\cd)$ is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums $G(\\k^\\la) (1\\laN-1)$ in index 2 case with order $N$ being general even integer (i.e. $N=2^{r}\\cd N_0$ where $r,N_0$ are positive integers and $N_03$ is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-09-28T08:34:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L05",
      "11L20"
    ],
    "keywords": [
      "gauss sums",
      "explicit evaluation",
      "complete solving",
      "positive integer",
      "prime number"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Science in China A: Mathematics",
      "year": 2010,
      "month": "Jun",
      "volume": 53,
      "number": 9,
      "pages": 2525
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010ScChA..53.2525Y"
    }
  }
}