In a real closed field, the formula $\exists z\,(z^2=x)$ defines the nonnegative elements. A quantifier-free formula in one variable over the prime field is a Boolean combination of polynomial equations, so in an infinite field it defines a finite or cofinite set. The nonnegative cone is neither finite nor cofinite. Hence the <Theory of real closed fields> does not eliminate quantifiers in the pure ring language.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 144 / 3 / i / Solution

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The theory of real closed fields is complete and model-complete. In the language of ordered rings it eliminates quantifiers; in the pure language of rings its definable order prevents quantifier elimination.


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 Incoming links: Theory of real closed fields