Abstract: A differential field F is a field with a derivation D; that is, an additive endomorphism satisfying D(ab)=aD(b)+bD(a). The constant field C of F is the kernel of D. C is assumed algeraically closed and characteristic zero. Let L=D^n+a_1D^{n-1}+ \dots a_nD^0 be a (linear, monic) differential operator over F. The (unique) field generated over F by a full set of solutions to the (homogeneous) differential equation L=0, and containing no new constants, is called the differential Galois, or Picard--Vessiot, extension of F for L. The compositum of all of these for all L is the Picard--Vessiot closure of F. Such closures may have proper differential Galois extensions, hence closures of their own. Taking these and iterating gives the complete Picard--Vessiot closure. We discuss what is known about this field, its group of differential authomorphisms, and the correspondence between the latter's subgroups and differential subfields of the complete closure.

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