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In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the language of rings (ordered rings) must be algebraically closed (real closed). I want to know if there is a similar result for ordered abelian groups.

I read in this thesis, http://www.flypig.co.uk/maths/Thesis.pdf, that Robinson proof in his book "Complete Theories" that a theory of ordered abelian groups has quantifier elimination if and only if it is the theory of ordered divisible abelian groups, but I don't see in the book of Robinson where he proof that statement. Can you help, thank you.

Requiem
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Robinson’s result is the following:

The model completion of the theory of totally ordered abelian groups T is the theory T’ of totally ordered divisible abelian groups with at least two different elements.

This statement you should be able to find.

This implies the statement you cite. Recall that for a theory T with a universal axiomatization, T’ having QE is equivalent to T’ being the model completion of T.