{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "00DW862T78"
    },
    {
      "field": "uses",
      "target": "085MJS6907"
    }
  ],
  "body": "\n## Statement\n\nLet $p$ be an odd prime.\nThen\n$$\\left(\\frac{-1}{p}\\right) = (-1)^{(p-1)/2},$$\nwhere the left side is a [[Legendre symbol|01J8AMV9RS]].\n\n## Remarks\n\nSo $\\left(\\frac{-1}{p}\\right) = 1$ if $p \\equiv 1 \\pmod 4$, and $\\left(\\frac{-1}{p}\\right) = -1$ if $p \\equiv 3 \\pmod 4$.\nThe proof is [[Euler's criterion|0M7DEDKSYB]] with $a = -1$.\nFor example, $2^2 = 4 \\equiv -1 \\pmod 5$, but $-1$ is not a square modulo $7$.\n\n## Explanations\n\n### L1 human\n\nThe number $-1$ is a square modulo an odd prime $p$ exactly when $p$ leaves remainder $1$ on division by $4$.\nSo $-1$ is a square modulo $5$, $13$ and $17$, but not modulo $3$, $7$ and $11$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "11A15"
    ],
    "name": "First supplement to quadratic reciprocity",
    "role": "theorem",
    "sources": [
      {
        "locator": "q = ±1 and the first supplement",
        "url": "https://en.wikipedia.org/wiki/Quadratic_reciprocity"
      }
    ],
    "tag": "0PHBS78HKT",
    "uses": [
      "01J8AMV9RS",
      "0M7DEDKSYB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0PHBS78HKT/",
  "links": [
    {
      "field": "uses",
      "target": "01J8AMV9RS"
    },
    {
      "field": "uses",
      "target": "0M7DEDKSYB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0PHBS78HKT.md",
  "msc": [
    "11A15"
  ],
  "name": "First supplement to quadratic reciprocity",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0PH/0PHBS78HKT.md",
  "status": null,
  "tag": "0PHBS78HKT",
  "uri": "m2://0PHBS78HKT",
  "url": "https://math2.tamnd.com/wiki/first-supplement-to-quadratic-reciprocity/",
  "writing": {
    "tag": "0PHBS78HKT",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}