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  "body": "\n## Summary\n\nA prime model of a complete theory is a model that embeds elementarily into every model of the theory.\nIt is the smallest model in a strong sense, and when it exists in a countable language it is unique up to isomorphism.\nFor a complete theory in a countable language, the prime models are exactly the countable atomic models, in which every tuple satisfies an isolated [[type|9V71Q3SD1G]].\nThe field of algebraic numbers is the prime model of the theory of algebraically closed fields of characteristic $0$.\n\n## Explanations\n\n### L1 human\n\nAmong all models of a theory, the prime model is the one that sits inside every other model, in a way that keeps all first-order truths.\nIt contains only the elements that the theory forces to exist.\nFor algebraically closed fields of characteristic $0$, these are the algebraic numbers: every such field contains a copy of them, and nothing smaller works.\n",
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