School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

University of Oklahoma

of a Differential Field

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.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006