Model complete theories without quantifier elimination

Question

As we know, if a theory $T$ admits quantifier elimination, then $T$ is model complete. What are the simplest examples that show that the converse is not true?

Answer 1

As Mauro indicated, I had already posted an example in a previous post. One counterexample is the theory of real closed fields in the language of rings $\{0,1,+,\cdot\}$.

It is easy to see that it does not have q.e.: the formula $(\exists x)x^2+y=z$ is not equivalent to a quantifier-free formula, as you can't tell whether or not one of two algebraically independent reals is greater than the other based on their atomic diagram: swapping them around leaves the diagram unchanged, as it just says that the two satisfy no nontrivial polynomial equations.

To see that it actually is model complete is tougher. Since $\leq$ is definable (by the formula from the previous paragraph), it is enough to show that the theory of real closed fields in the language $\{0,1,+,\cdot,\leq\}$ has quantifier elimination. This is classical (AFAIK due to Tarski), and can be found in many places, for instance on page 4 (43) of these lecture notes by Marker.