A feature of algebraically closed fields, real closed fields and pseudofinite fields is that they have only finitely many extensions of degree \(n\) for every positive integer \(n\). The authors point out that this is due to a common model-theoretic property, based on the following notion of dimension. Let \(M\) be any first-order structure, \(U\) and \(V\) be (parametrically) definable sets (in \(M^{\text{eq}})\). Then say
(i) \(\dim (U)\leq \dim (V)\) if and only if there are a partition of \(U\) into finitely many definable subsets \(U_0, \dots,U_m\), and definable functions with finite fibres \(f_0, \dots, f_m\) from \(U_0, \dots, U_m\) respectively in \(V\);
(ii) \(\dim (U)= \dim (V)\) if and only if both \(\dim (U)\leq \dim(V)\) and \(\dim (V)\leq \dim (U)\).
A (very saturated) structure \(M\) is called surgical if and only if, whenever \(U\) is a definable set and \(E\) is a definable equivalence relation in \(U\), then only finitely many \(E\)-classes have the same dimension as \(U\). Surgical structures include totally transcendental structures (so algebraically closed fields), \(o\)-minimal structures (in particular real closed fields), and pseudofinite fields. The authors show that every surgical field is perfect and has only finitely many extensions of degree \(n\) for every \(n\). The proof uses the following weakened form of surgery: for every definable action of a group \(G\) on a set \(U\), only finitely many orbits have the same dimension as \(U\).
A final comment. The paper is written in three languages: English, Esperanto and (for the most part) French. It might be interesting to know if this is intended as a contribution to universal peace, or as the affirmation of a superiority of French civilization.