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  "body": "\n## Summary\n\nA real closure of an [[ordered field|0P3VCBDEE2]] is a real closed algebraic extension whose order extends the given one.\nA real closed field is a [[formally real field|X32TSWJSYT]] with no proper formally real algebraic extension, such as $\\R$.\nEvery ordered field has a real closure, and it is unique up to a unique isomorphism over the field.\nIt plays for ordered fields the role that the [[algebraic closure|0X6AB5P3G1]] plays for fields.\n\n## Explanations\n\n### L1 human\n\nThe real numbers have the property that every positive number has a square root and every odd-degree polynomial has a root.\nThe real closure of an ordered field adds exactly what is needed to get these properties, and nothing more.\nThe real closure of $\\Q$ is the field of real algebraic numbers, such as $\\sqrt 2$ and the real cube root of $5$, but not $\\pi$.\n",
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