# 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

Hint: Marker is offering this as a corollary to his Theorem 3.1.9 which states that the theory DAG of torsion-free divisible abelian groups has quantifier elimination. It isn't really a corollary, but rather a generalisation: Marker has tried to help in the text before the corollary by identifying the lemmas that prove (i) and (ii) in the special case of DAG, but is expecting you to figure out how these lemmas are used in the proof of Theorem 3.1.9. He has made this a bit harder for you by only referring explicitly to one of the two lemmas in that proof.

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 $G \models \exists y\phi(\overline{G}, y)$.