 Solution

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.