Automorphism groups of differentially closed fields Original Research Article

We examine the connections between several automorphism groups associated with a saturated differentially closed field U of characteristic zero. These groups are: Γ, the automorphism group of U; the automorphism group of Γ; View the MathML source, the automorphism group of the differential combinatorial geometry of U and View the MathML source, the group of field automorphisms of U that respect differential closure. Our main results are: • If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then the set of subgroups of Γ consisting of all pointwise stabilizers of Dclʹs of finite subsets of U is invariant under Aut(Γ) (Theorem 2.10). • If U is of arbitrary infinite cardinality, then each automorphism of the differential combinatorial geometry is induced by a field automorphism of U that respects differential closure (Theorem 3.1). • If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then each automorphism of Γ is induced by an element of View the MathML source acting on Γ by conjugation (Theorem 4.10). • If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then the outer automorphism group of Γ is isomorphic to the multiplicative group of the rationals (Theorem 4.18).