A model of an elementary theory $T$ that can be embedded in every model of $T$. If $T$ is model complete (cf. Model theory) and admits a prime model, then $T$ is complete. This fact is called the prime model test.
The prime model of the elementary theory of all fields with fixed characteristic is the prime field of that characteristic.
Prime model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_model&oldid=31517