I want to show that if a model $M$ is minimal and $\omega$-saturated, then the Theory($M$) is strongly minimal.
I know that the reverse statement would be trivial. However, I'm not sure about the forward statement. More specifically, how is it that just the minimality and $\omega$-saturatedeness of one model result in the whole theory being strongly minimal.
Hint: If $T = \text{Th}(M)$ is not strongly minimal, then there is some model $N\models T$ and some formula $\varphi(x;\bar{b})$ with $\bar{b}$ from $N$ such that $\varphi(N;\bar{b})$ is infinite and coinfinite. Now use the fact that $M$ is $\omega$-saturated to find such a witness in $M$ (it is enough to realize $\text{tp}(\bar{b}/\emptyset)$ in $M$!), contradicting minimality of $M$.