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  "body": "\n## Statement\n\nEvery [[field|0QA8CHH8MG]] $K$ has an [[algebraic closure|0MCXX29HWY]].\n\n## Remarks\n\nOne proof: take a polynomial ring $R$ over $K$ with one variable for each root of each monic irreducible polynomial over $K$. A [[maximal ideal|02HJYT53F7]] of $R$ that contains the relations gives a field $K_1$ in which every polynomial over $K$ splits.\nRepeating the step gives fields $K \\subseteq K_1 \\subseteq K_2 \\subseteq \\dots$, and their union is an algebraic closure of $K$.\nThe proof uses Zorn's lemma, through the existence of maximal ideals. The weaker ultrafilter lemma is enough.\nFor $K = \\mathbb{Q}$ or $K = \\mathbb{R}$, the complex numbers give a direct proof.\n\n## Explanations\n\n### L1 human\n\nEvery field sits inside an algebraically closed field that is algebraic over it.\nThe general proof needs a form of the axiom of choice, because a field can have very many polynomials.\n",
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