Automorphism groups of differentially closed fields
A differentially closed field (dcf) is an algebraically closed differential field in which for every two differential polynomials (polynomials in several variables and their derivatives) \(f\) and \(g\) such that the order of \(f\) is greater than the order of \(g\), there is an \(x\) such that \(f(x)=0\) but \(g(x)\not= 0\). The theory of (set of first-order formulas that hold true in every) dcf of characteristic zero (in the language which extends the language of rings by a function-symbol for the derivative) is complete and model complete (every formula is equivalent to an existential one). In some respects this theory is similar to that of algebraically closed fields (acf). The paper proves a number of results for the automorphism group of a dcf, some similar to and others different than analogous results (e.g. of Evans and Lascar) for acf. A saturated dcf is a notion analogous to that of an acf of infinite transcendence degree (at least in the countable case). The paper proves a number of results for the automorphisms Aut\((U)\) of a saturated dcf \(U\), a central result being the following: A saturated dcf \(U\) of characteristic zero and cardinality \(2^{\lambda}=\lambda ^{+}\) has outer automorphisms and the outer automorphism group of Aut\((U)\) is isomorphic to the multiplicative group of the rationals.
scientific article; zbMATH DE number 1838865
scientific article