{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $K$ be a [[field|0QA8CHH8MG]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $K$.\n2. Since $I$ is an additive subgroup of $K$, $0 \\in I$. So if $I$ has no nonzero element, then $I = \\set{0}$.\n3. Otherwise, let $a \\in I$ with $a \\neq 0$. Since $K$ is a field, there is $b \\in K$ with $b a = 1$.\n4. Since $I$ is an ideal, $b a \\in I$. So $1 \\in I$.\n5. Let $r \\in K$. Since $I$ is an ideal and $1 \\in I$, $r = r \\cdot 1 \\in I$. So $K \\subseteq I$, and $I = K$.\n6. Hence $I = \\set{0}$ or $I = K$.\n\n## Idea\n\nA nonzero element of an ideal of a field can be multiplied by its inverse. So the ideal contains $1$, and then it contains every multiple of $1$, which is every element.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "12E99"
    ],
    "name": "Proof that the only ideals of a field are zero and the whole field",
    "proves": "0M6G75E95E",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Ideals_of_Field"
      }
    ],
    "tag": "HAP2G0H9NC",
    "uses": [
      "0QA8CHH8MG",
      "0MYSA6MYNA"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/HAP2G0H9NC/",
  "links": [
    {
      "field": "proves",
      "target": "0M6G75E95E"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "0MYSA6MYNA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HAP2G0H9NC.md",
  "msc": [
    "12E99"
  ],
  "name": "Proof that the only ideals of a field are zero and the whole field",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HAP/HAP2G0H9NC.md",
  "status": null,
  "tag": "HAP2G0H9NC",
  "uri": "m2://HAP2G0H9NC",
  "url": "https://math2.tamnd.com/wiki/proof-that-the-only-ideals-of-a-field-are-zero-and-the-whole-field/",
  "writing": {
    "tag": "HAP2G0H9NC",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}