Published online by Cambridge University Press: 12 March 2014
If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.
The existence of prime models was proved by Morley  and their uniqueness for countable A by Vaught . Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this follows
If T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by .)
By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum  for the total transcendency of the theory of differentially closed fields.
We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).
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