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Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, Trans. AMS 351, 2997–3071. G. Cherlin, Algebraically closed commutative rings, J. Symbolic Logic 38 (1973), 493–499. E. Compoint and M. Singer, Computing Galois groups of completely reducible differential equations, J. Symbolic Comput. 28 (1999), 473–494. F. Delon, Id´eaux et types sur les corps s´eparablement clos, Suppl´ement au Bull. SMF, M´emoire 33, Tome 116 (1988). L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields.

4) Universal domains: the functor in (3) is replaced by its value at a single structure; this can work only when a single structure can be viewed as the amalgam of “all structures”. Approach (4) is possible only when the class of structures in question admits amalgamation. When it does, (4) is entirely equivalent to (3); any structure admits an embedding in the universal domain, unique up to an automorphism of the universal domain; permitting recovery of the value of the functor there. The history of algebra passed through (1) (complex numbers in the 16th century), (4) (complex numbers in the 19th), (2) (“modern algebra”), (4) (Weil’s universal domains, in any characteristic), (1) + (3) (Grothendieck).

Proof. 3 for V, Vp respectively. 30 , it suffices to show that Q⊗∆ = Q⊗∆(p) for ad-almost all p. 12. 1. 1 (any troublesome ingredients having been dealt with). One first considers the uniform cover H(V ) = Aut(V /k(t))F of the Galois group of V , and shows that H(V )p = H(Vp ) for ad-almost all p. 8. Thus also (H(V )t )p = H(Vp )t for ad-almost all p. 3, let Gt (V ) be the image of Aut(V /k(t))0 in H(V )/H(V )t . It will suffice to show that Gt (V )p = Gt (Vp ) for gr-almost all p. 10 does the job.

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