For homework, i need to prove the following: Let $M$ be a structure with universe $\mathbb {Q}$ of the language that cosiest only of function symbols $f_{\bar {\alpha}}(\bar{x})$ for all $\bar {\alpha}=\{\alpha_i \mid i<n\}\in\mathbb{Q}^n$ for some $n$ s.t $\sum_i \alpha_i =1$. and the functions are interpreted by $f_{\bar{\alpha}}(\bar{x})=\sum_i \alpha_i x_i$
Define $T=Th(M)$. In the first section I need to show that $T$ is strongly minimal, I think i managed to do that by quantifier elimination. since then every atomic formula has one realisations or co-finte set of solutions, and then by induction we can easily see that all formulas are finite or co-finite.
For the Second section in which i need help, is to show that $M$ is a prime model of $T$ and that $dim(M)=1$
Where by dimension, i mean the size of a basis of the pre-geometry defined on $M$ by the algebraic closer. $acl(\cdot)$
For the prime model part, I think maybe we can show that every element of $\mathbb{Q}$ is definable in $T$ by some linear function?
And for the dimension part I am quite lost, I know we need to find two independent and spanning elements in $\mathbb{Q}$ but I'm not sure what does it mean here.
Edit: I was given a clue, fix a number as zero, and define a structure of a vector space above $\mathbb {Q}$. I see how i can do that with the functions in out language. and i see that the prime model is indeed $M$ and the dimension is 1 as a vector space. But i am not sure why is it corresponding with the dimension of the pre-geomtry. And i was wondering if there is a way to prove section one in a similar way.