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  "body": "\n## Definition\n\nA field $R$ is *real closed* if $R$ is [[formally real|X32TSWJSYT]] and no [[algebraic extension|09EYMKMSB9]] $R' \\ne R$ of $R$ is formally real.\nLet $(F, \\le)$ be an [[ordered field|010JCV9J4E]].\nA *real closure* of $(F, \\le)$ is an algebraic extension $R$ of $F$ such that:\n\n1. $R$ is real closed;\n2. the ordering of $R$, in which the nonnegative elements are the squares, restricts to $\\le$ on $F$.\n\n## Remarks\n\nA real closed field has exactly one ordering, given by $a \\ge 0$ if and only if $a$ is a square, so clause 2 makes sense.\nEquivalently, $R$ is real closed if and only if two things hold.\nIt has an ordering in which every positive element is a square, and every polynomial of odd degree over $R$ has a root in $R$.\nEvery ordered field has a real closure, and it is unique up to a unique isomorphism over $F$ that respects the orderings.\nExample: the field of real [[algebraic numbers|0MB4TGMXG6]] is a real closure of $\\Q$.\nNon-example: $\\R$ is real closed, but it is not a real closure of $\\Q$, because it is not algebraic over $\\Q$.\n",
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