{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\alpha_1, \\dots, \\alpha_n$ be distinct [[algebraic numbers|0MB4TGMXG6]].\nThen $e^{\\alpha_1}, \\dots, e^{\\alpha_n}$ are linearly independent over the field of algebraic numbers.\nThat is, if $\\beta_1, \\dots, \\beta_n$ are algebraic numbers with $\\beta_1 e^{\\alpha_1} + \\dots + \\beta_n e^{\\alpha_n} = 0$, then $\\beta_1 = \\dots = \\beta_n = 0$.\n\n## Remarks\n\nA consequence: $e^{\\alpha}$ is [[transcendental|079EGSD32Y]] for each nonzero algebraic number $\\alpha$. In particular, $e$ is transcendental.\nAnother consequence: $\\pi$ is transcendental. Otherwise $i\\pi$ would be algebraic, and $e^{i\\pi} + 1 = 0$ would contradict the theorem.\nSo the circle cannot be squared with ruler and compass.\n\n## History\n\nFerdinand von Lindemann proved in 1882 that $e^{\\alpha}$ is transcendental for each nonzero algebraic $\\alpha$.\nKarl Weierstrass proved the general statement in 1885.\n\n## Explanations\n\n### L1 human\n\nThe exponential function sends nonzero algebraic numbers to transcendental numbers.\nWith $\\alpha = 1$ this shows that $e$ is transcendental, and with Euler's formula it shows the same for $\\pi$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "11J81"
    ],
    "name": "Lindemann–Weierstrass theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1572474",
      "wikipedia": "Lindemann–Weierstrass theorem"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem"
      }
    ],
    "tag": "0JNCBFMPXR",
    "uses": [
      "0MB4TGMXG6",
      "079EGSD32Y"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0JNCBFMPXR/",
  "links": [
    {
      "field": "uses",
      "target": "0MB4TGMXG6"
    },
    {
      "field": "uses",
      "target": "079EGSD32Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JNCBFMPXR.md",
  "msc": [
    "11J81"
  ],
  "name": "Lindemann–Weierstrass theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JN/0JNCBFMPXR.md",
  "status": null,
  "tag": "0JNCBFMPXR",
  "uri": "m2://0JNCBFMPXR",
  "url": "https://math2.tamnd.com/wiki/lindemann-weierstrass-theorem/",
  "writing": {
    "tag": "0JNCBFMPXR",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}