---
tag: S8W40PYNHX
kind: concept
name: Prime model
msc: [03C50, 03C15]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_model
same_as:
  wikidata: Q7243376
  wikipedia: 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|9V71Q3SD1G]].
The field of algebraic numbers is the prime model of the theory of algebraically closed fields of characteristic $0$.

## Explanations

### L1 human

Among 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.
It contains only the elements that the theory forces to exist.
For algebraically closed fields of characteristic $0$, these are the algebraic numbers: every such field contains a copy of them, and nothing smaller works.