A note on the axioms for differentially closed fields of characteristic zero
All fields dealt with in the paper are of characteristic zero. A field \(k\) equipped with an additive homomorphism \(D:k\rightarrow k\) such that \(D(a\cdot b)=D(a)\cdot b+a\cdot D(b)\) is called a differential field with a derivation \(D\). In [Differentially closed fields: A model-theoretic tour, in ``Contributions to Algebra'', 37-61 (1977; Zbl 0368.12013)], \textit{L. Blum} found axioms for \(DCF_0\), the theory of differentially closed fields of characteristic zero. If \(V\) is a variety defined over the field of constants \(C_k\) of \(k\), then for \(a\in V(k),\) the derivative \(D(a)\) naturally lives in the tangent space to \(V\) at \(a,\) thus \(D\) determines a section of the tangent bundle \(T(V)\). If \(V\) is not necessarily defined over the constants of \(k\), then \(D\) determines a section of a certain torsor \(\tau (V)\) under \(T(V)\). The authors axiomatize \(DCF_0\) and in a manner which is suggested by this geometric interpretaion. Two axioms for \(DCF_0\) are formulated. As a derivation \(D\) on a field \(k\) has a unique extension to a derivation on the algebraic closure of \(k,\) it is sufficient to deal with an algebraically closed field \(k.\) (i) The field \(k\) is algebraically closed. (ii) Let \(V\subset K^n\) and \(W\subset \tau(V)\) be irreducible varieties defined over \(k\) such that \(W\) projects generically on \(V\) (that is, a generic point of \(W\) over \(k\) is of the form \((\overline{a},\overline{b})\), where \(\overline{a}\) is a generic point of \(V\) over \(k\)). Let \(U\) be a nonempty Zariski-open subset of \(W\) defined over \(k\). Then there is a point \((\overline{a},\overline{b})\in U(k)\) such that \(\overline{b}=D(\overline{a})\). Finally some basic facts about differentially closed fields (\(\omega\)-stability and DCC on differential varieties) are proved without going through classical differential algebra.
scientific article; zbMATH DE number 1218690