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  "body": "\n## Definition\n\nLet $L/K$ be an [[algebraic extension|09EYMKMSB9]].\nThe extension $L/K$ is normal if every [[irreducible polynomial|00X2A9G9W3]] $f \\in K[x]$ that has a root in $L$ [[splits|0SHMC00RJV]] over $L$.\n\n## Remarks\n\nA finite extension $L/K$ is normal if and only if $L$ is the [[splitting field|02NZZ1J0FH]] of some polynomial over $K$.\nEvery extension of degree $2$ is normal. For example, $\\mathbb{Q}(\\sqrt{2})/\\mathbb{Q}$ is normal.\nThe extension $\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q}$ is not normal: $x^3 - 2$ has a root in it, but its other two roots are not real.\nNormality is not transitive: $\\mathbb{Q}(\\sqrt[4]{2})/\\mathbb{Q}(\\sqrt{2})$ and $\\mathbb{Q}(\\sqrt{2})/\\mathbb{Q}$ are normal, but $\\mathbb{Q}(\\sqrt[4]{2})/\\mathbb{Q}$ is not.\n\n## Explanations\n\n### L1 human\n\nA normal extension contains all the roots of a polynomial as soon as it contains one of them.\nIt has no lopsided gaps, like a field that contains one cube root of $2$ but not the others.\n",
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