# A test for quantifier eliumination

In David Marker's "Model Theory: An Introduction" book I was trying to prove Corollary 3.1.12 which is left for the reader, but I couldn't reach any solution. The aforementioned corollary states:

Suppose that $T$ is an $\mathcal{L}-$theory such that:

$i)$ $T$ has algebraically prime models, and

$ii)$ $\mathcal{M}\prec_{s}\mathcal{N}$ whenever $\mathcal{M}\subseteq\mathcal{N}$ are models of T;

Then $T$ has quantifier elimination.

Would be thankful for your help.

• I found the proof here on page 13. The author notes that the "proof is very long" and "[he] has yet to internalize it". Maybe someone else has a shorter proof Jan 14 '16 at 22:57

The idea is that (i) generalises the existence of divisible hulls and (ii) generalises the property that if $G \subseteq H$ is an inclusion of DAGs and $H \models \exists y\phi(\overline{g}, y)$, where $\phi(\overline{x}, y)$ is quantifier-free and $\overline{g} \in G$, then $G \models \exists y\phi(\overline{G}, y)$.