We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in sans-serifCODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X ⊆ M n , there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

We prove that if T is a theory of large, bounded, fields of characteristic 0 with almost quantifier elimination, and T D is the model companion of T ∪ {"∂ is a derivation"}, then for any model (U , ∂) of T D , differential subfield K of U such that C K |= T , and linear differential equation ∂Y = AY over K, there is a Picard-Vessiot extension L of K for the equation with K ≤ L ≤ U , i.e. L can be embedded in U over K, as a differential field. Moreover such L is unique to isomorphism over K as a differential field. Likewise for the analogue for strongly normal extensions for logarithmic differential equations in the sense of Kolchin.Recent papers such as [2] and [5] have shown that under certain conditions (on the differential field (K, ∂) and its field C K of constants), given a linear differential equation over (K, ∂) we can find a Picard-Vessiot extension (L, ∂) of (K, ∂) for the equation such that C K is existentially closed in L (as a field). Among the motivating examples to which this applies is the case where C K is * 2. A field K is said to be large (or ample) if for any algebraic variety V which is defined over K, is K-irreducible, and has a nonsingular K-rational point, V (K) is Zariski dense in V .Remark 1.2. 1. Large fields were introduced by Pop in [13] where one can find other characterizations.2. The class of large fields is elementary in the language L.

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We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois' nullstellensatz and Stengle's positivstellensatz for ordered fields.