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  "body": "\n## Statement\n\nLet $\\mathcal{L} = \\set{+, -, \\cdot, 0, 1, \u003c}$ be the language of ordered rings.\nLet $\\mathrm{RCF}$ be the $\\mathcal{L}$-[[theory|92SWXST1ZM]] of [[real closed fields|DDY3Q3PJ6W]] with their unique field ordering.\nIts axioms are those of an [[ordered field|0P3VCBDEE2]], together with two more.\nOne axiom says that every positive element has a square root.\nOne scheme says that every polynomial of odd degree has a root.\nThen $\\mathrm{RCF}$ admits [[quantifier elimination|7FDDXV3JK7]]: for every $\\mathcal{L}$-formula $\\varphi(x_1, \\dots, x_n)$ there is a quantifier-free $\\mathcal{L}$-formula $\\psi(x_1, \\dots, x_n)$ such that\n$$\\mathrm{RCF} \\vdash \\forall x_1 \\cdots \\forall x_n \\, \\bigl(\\varphi(x_1, \\dots, x_n) \\leftrightarrow \\psi(x_1, \\dots, x_n)\\bigr).$$\n\n## Remarks\n\nThe theorem is due to Tarski.\nIt suffices to eliminate one existential quantifier in front of a conjunction of polynomial equations and inequalities in one variable $x$.\nSuch a conjunction has a solution exactly when certain sign conditions on the coefficients hold.\nThese conditions can be computed, for instance with Sturm sequences and the intermediate value property of polynomials over a real closed field.\n\nExample: for real closed fields,\n$$\\exists x \\, (x^2 + b x + c = 0) \\quad \\leftrightarrow \\quad b^2 - 4c \u003e 0 \\ \\lor \\ b^2 - 4c = 0.$$\n\nConsequences: $\\mathrm{RCF}$ is [[complete|655K8Y7VS8]], so every real closed field satisfies the same first-order sentences as $\\R$; and $\\mathrm{RCF}$ is decidable.\nGeometrically, the projection of a semialgebraic set is semialgebraic, which is the [[Tarski–Seidenberg theorem|3W5JGDZAN0]].\n\nThe symbol $\u003c$ is necessary.\nIn the language of rings alone, the set $\\set{x \\in \\R : \\exists y \\, (x = y^2)} = [0, \\infty)$ is definable, but every quantifier-free definable subset of $\\R$ in that language is finite or cofinite.\n\n## Explanations\n\n### L1 human\n\nA statement about real numbers that uses \"there exists\" and \"for all\", together with addition, multiplication and order, can be rewritten without \"there exists\" and \"for all\".\nWhat remains is a combination of polynomial equations and inequalities in the free variables.\nFor example, \"the quadratic $x^2 + bx + c$ has a real root\" is the same as \"$b^2 - 4c \\ge 0$\".\nSo every such question about the real numbers can, in principle, be decided by a computation.\n\n### L2 human\n\nQuantifier elimination is the model-theoretic core of real algebraic geometry.\nIt shows that the sets definable in $\\R$ with $+$, $\\cdot$ and $\u003c$ are exactly the semialgebraic sets, and that projections and images under polynomial maps stay in this class.\nIt gives the completeness of the theory of real closed fields, which transfers results from $\\R$ to every real closed field.\nThis transfer gives a short modern proof of Artin's solution of [[Hilbert's seventeenth problem|J45F4F13VB]].\nIt also shows that the theory of $\\R$ is [[o-minimal|DTWHAW1N0V]]: every definable subset of the line is a finite union of points and intervals.\nThe analogous result for [[algebraically closed fields|MT8TDVPBXB]] uses only the language of rings.\n",
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