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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $K$.\nThen $I = \\{0\\}$ or $I = K$.\n\n## Remarks\n\nThe proof: if $a \\in I$ and $a \\neq 0$, then $1 = a^{-1} a \\in I$, so $r = r \\cdot 1 \\in I$ for each $r \\in K$.\nThe converse holds: a nonzero commutative ring whose only ideals are $\\{0\\}$ and the whole ring is a field.\nFor each nonzero $a$, the ideal $(a)$ is the whole ring, so $1 = b a$ for some $b$.\n\n## Explanations\n\n### L1 human\n\nIf an ideal of a field contains a nonzero element, then it contains $1$, because you can divide.\nOnce an ideal contains $1$, it contains every element.\nSo a field has only the two trivial ideals.\n",
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  "name": "The only ideals of a field are zero and the whole field",
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