On differential Galois groups of strongly normal extensions

We give a detailed proof of Kolchins results on differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. We closely follow former works due to Pillay and his co-authors which were written under the assumption that the field of constant is algebraically closed. In the present setting, which encompasses the cases of 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 the theory of closed ordered fields, we establish a relative Galois correspondence for definable subgroups of the group of differential order automorphisms.