Prime model

Summary

A prime model of a complete theory is a model that embeds elementarily into every model of the theory. It is the smallest model in a strong sense, and when it exists in a countable language it is unique up to isomorphism. For a complete theory in a countable language, the prime models are exactly the countable atomic models, in which every tuple satisfies an isolated type. The field of algebraic numbers is the prime model of the theory of algebraically closed fields of characteristic $0$.