Intersections of algebraically closed fields (Q1076073)

In this well written paper the authors prove at first the equality \[ \mathcal D_L(F\cap G)=\mathcal L_L(\mathcal D_L(F)\cup \mathcal D_L(G)),\] where \(\mathcal D_L(H)\) is the set of derivations of a field \(L\) that kill a field \(H\subseteq L\) and \(\mathcal L_L(X)\) denotes the Lie subalgebra of \(\mathcal D= \mathcal D_ L(K)\) \((K\) is a base field) generated by \(X\). Using this theorem they show that the Nerode-Remmel hypothesis holds [\textit{A. Nerode} and \textit{J. Remmel}, Patras Logic Symposion, Proc., Patras/Greece 1980, Stud. Logic Found. Math. 109, 41--65 (1982; Zbl 0526.03026)], namely, there is a recursive function which, given canonical indices of finite bases of \(F\) and \(G\), computes \(\operatorname{tr}\deg(F\cap G)\). Further the authors say that the fields \(F\) and \(G\) intersect maximally if \[ \operatorname{tr}\deg (F\cap G) = \operatorname{tr}\deg F + \operatorname{tr}\deg G - \operatorname{tr}\deg (F\vee G). \] They show that this is equivalent to ``the vector space generated by \(\mathcal D_ L(F)\cup \mathcal D_ L(G)\) is a Lie algebra'', and they obtain a further characterization of maximal intersections.