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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nAn algebraic closure of $K$ is a [[field extension|0WQ1A8SHTZ]] $\\overline{K}/K$ such that $\\overline{K}/K$ is [[algebraic|09EYMKMSB9]] and $\\overline{K}$ is [[algebraically closed|02SJG1HC32]].\n\n## Remarks\n\nThe field $\\mathbb{C}$ is an algebraic closure of $\\mathbb{R}$, by the fundamental theorem of algebra.\nThe field $\\overline{\\mathbb{Q}}$ of [[algebraic numbers|0QFTETCHKB]] is an algebraic closure of $\\mathbb{Q}$. The field $\\mathbb{C}$ is not, because $\\mathbb{C}/\\mathbb{Q}$ is not algebraic.\nIf $K$ is infinite, then $\\overline{K}$ has the same cardinality as $K$. If $K$ is finite, then $\\overline{K}$ is countably infinite.\nAn algebraic closure is a largest algebraic extension of $K$, and a smallest algebraically closed field that contains $K$.\n\n## Explanations\n\n### L1 human\n\nThe algebraic closure adds to a field all the roots of all its polynomials, and nothing else.\nFor the rationals, it gives the algebraic numbers, not all the complex numbers.\n",
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