A model P of a complete theory T is prime if P elementarily embeds into every model of T; it is a minimal (with respect to elementary embeddings) canonical representative of the theory.

Principle

Minimality by elementary embedding: primality selects a smallest model that must appear inside any other model of the same complete theory, serving as a base or atomic core from which others extend.

Demonstration

For the theory of algebraically closed fields of fixed characteristic, the algebraic closure of the prime field (e.g., the algebraic closure of Q or of F_p) is a prime model: it embeds elementarily into every algebraically closed field of that characteristic.

Misapplication

Equating prime with smallest cardinality or assuming every theory has a prime model; a prime model is minimal in the sense of embeddings, not necessarily of cardinality, and many theories admit no prime model at all.

Consequence

When it exists, a prime model provides a canonical minimal structure that often is atomic and is useful for constructing and comparing models; prime models help analyze definable closures and ground-level algebraicity properties.

Reversal

A theory without a prime model lacks a canonical minimal embedding-base; models then may be mutually incomparable by elementary embedding, and classification requires other invariants like saturated or minimal types.

Boundary

Prime models are defined for complete first-order theories and require existence proofs; existence can fail for incomplete theories or those with too few isolated types, and primality depends on the chosen language and completeness assumption.

Semantic Tension

Related to atomic models and to minimal models in algebraic contexts: prime models are often atomic in countable complete theories, but primality emphasizes embedding-universality while atomicity emphasizes isolated types, producing a subtle distinction.

Synthesis

A prime model is the minimal elementary building block of a complete theory: it embeds into every model of the theory and, when present, gives a canonical, often atomic, seed from which other models are constructed or compared.