{
  "id": "1807.01181",
  "version": "v1",
  "published": "2018-07-02T15:36:05.000Z",
  "updated": "2018-07-02T15:36:05.000Z",
  "title": "On sums and products in a field",
  "authors": [
    "Guang-Liang Zhou",
    "Zhi-Wei Sun"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we study sums and products in a field. Let $F$ be a field with ${\\rm ch}(F)\\not=2$, where ${\\rm ch}(F)$ is the characteristic of $F$. For any integer $k\\ge4$, we show that each $x\\in F$ can be written as $a_1+\\ldots+a_k$ with $a_1,\\ldots,a_k\\in F$ and $a_1\\ldots a_k=1$ if ${\\rm ch}(F)\\not=3$, and that for any $\\alpha\\in F\\setminus\\{0\\}$ we can write each $x\\in F$ as $a_1\\ldots a_k$ with $a_1,\\ldots,a_k\\in F$ and $a_1+\\ldots+a_k=\\alpha$. We also prove that for any $x\\in F$ and $k\\in\\{2,3,\\ldots\\}$ there are $a_1,\\ldots,a_{2k}\\in F$ such that $a_1+\\ldots+a_{2k}=x=a_1\\ldots a_{2k}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-02T15:36:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D85",
      "11P99",
      "11T99"
    ],
    "keywords": [
      "study sums",
      "characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}