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  "body": "\n## Example\n\nA *real closed field* is an [[ordered field|0P3VCBDEE2]] $(R, \u003c)$ with two properties.\nEvery positive element has a square root in $R$.\nEvery polynomial of odd degree with coefficients in $R$ has a root in $R$.\nThe [[real numbers|08J2M1JYAG]] form a real closed field.\nA positive real number has a real square root.\nA real polynomial of odd degree takes values of both signs, so it has a root by the intermediate value theorem.\nThe real algebraic numbers also form a real closed field, and it is countable.\n\nThe [[rational numbers|0NCJ5WXN6Z]] do not form a real closed field: $2 \u003e 0$ has no rational square root.\n\n## Remarks\n\nThe theory $\\mathrm{RCF}$ of real closed fields in the signature $\\set{+, -, \\cdot, 0, 1, \u003c}$ has [[quantifier elimination|FSJ9NGRRPA]].\nSo it is model complete, it is [[complete|655K8Y7VS8]], and it is decidable.\nEvery real closed field therefore satisfies the same first-order sentences as $\\R$.\nIn the signature of rings without $\u003c$, the theory is still model complete, but it no longer has quantifier elimination, since the order $x \\ge 0$ is defined by $\\exists y \\, (x = y^2)$.\n\nA field $R$ is real closed, for some ordering, exactly when $-1$ is not a sum of squares in $R$ and $R(\\sqrt{-1})$ is algebraically closed.\nThe ordering is then unique: the positive elements are the nonzero squares.\n\n## Explanations\n\n### L1 human\n\nReal closed fields are the fields that behave like the real numbers for all algebraic purposes.\nEvery positive number has a square root and odd-degree polynomials always have a root.\nTake any statement written with $+$, $\\cdot$, $\u003c$ and quantifiers over the elements of the field.\nModel theory shows that it is true in a real closed field exactly when it is true in $\\R$.\n",
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