Prime model

Remarks

Example: the field $\overline{\Q}$ of algebraic numbers is a prime model of $\mathrm{ACF}_0$, the theory of algebraically closed fields of characteristic $0$. Every algebraically closed field $K$ of characteristic $0$ contains $\Q$, and hence a copy of the algebraic closure of $\Q$. This field embedding is elementary, because $\mathrm{ACF}_0$ has quantifier elimination.

Non-example: $\C$ is a model of $\mathrm{ACF}_0$, but it is not prime, since it is uncountable and so cannot embed into the countable model $\overline{\Q}$.

If $L$ is countable, a prime model of $T$ is countable by the downward Löwenheim–Skolem theorem. Then a model is prime if and only if it is countable and atomic, that is, every tuple of it realizes an isolated type. Two prime models of $T$ are isomorphic. Another example is $(\Q, <)$, the prime model of the theory of dense linear orders without endpoints.