Model-theoretic proof of Hilbert's seventeenth problem
= Model-theoretic proof of Hilbert's seventeenth problem
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Model completeness of real closed fields transfers positivity of a rational function from $\mathbb R$ to real closed extensions of $\mathbb R$. If the function were not a sum of squares, the Artin-Schreier ordering criterion would produce a real closure in which it is negative, giving a contradiction.