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I know that for the complete theory $Th((\mathbb{Q},+,0))$ we have the prime model $(\mathbb{Q},+,0)$ and the countable saturated model $(\mathbb{Q}^{\infty},+,0)$, but what should I do when I try to prove this claim? Do I have to use the Ehrenfeuch-Fraisse Method?

Look up to the complete theory $Th((\mathbb{Q},<,0,1))$. I have to prove that there exists five 1-types of this theory, but that proves only that there are at least five 1-types, but how I have to prove that there exists precisly five 1-types?! I think I have to pick a arbitrary model of the theory and have to embed this into $(\mathbb{Q},<,0,1)$. Is this true? Can someone help me?

Thank you :)

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Note that any model of the theory is a $\bf Q$-vector space, as well as the other way around. Vector spaces are rather tame animals, you can just apply some basic linear algebra, or prove quantifier elimination. Ehrenfeucht-Fraisse method would not work too well here (at least not in its basic form), as the language is not a relational language.

As for your second question, it has already been asked here.

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