Real closed field

Remarks

The theory $\mathrm{RCF}$ of real closed fields in the signature $\set{+, -, \cdot, 0, 1, <}$ has quantifier elimination. So it is model complete, it is complete, and it is decidable. Every real closed field therefore satisfies the same first-order sentences as $\R$. In the signature of rings without $<$, the theory is still model complete, but it no longer has quantifier elimination, since the order $x \ge 0$ is defined by $\exists y \, (x = y^2)$.

A 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. The ordering is then unique: the positive elements are the nonzero squares.