An English Translation of Corps Et Chirurgie (fields and Surgery) by Anand Pillay & Bruno Poizat

Algebraically closed fields, real closed fields, and pseudofinite fields have, for every natural number n, have a finite number of extensions of degree n; we show they share this property with all fields which, like them, satisfy a very basic property of preservation of a type of model-theoretic dimension. This result is attained by showing that a certain action of the group GLn on such a field has only a finite number of orbits. The original article appeared in The Journal of Symbolic Logic, Volume 60, Number 2, June 1995 [4]. We have attempted to provide a translation as faithful to the original wording as possible. The one exception is in the final remark where we elaborate slightly for the sake of exposition. Any mistakes, typos, or inconsistencies are likely due to the translator and should be brought to their attention by emailing them at [email protected]. Dimension is the property that is conserved when one cuts something apart and puts the pieces back together or, similarly, if we replace one piece with two of its copies. This philosophy is what leads us to make the following definition: Definition 1. If U and V are two definable sets in a structure M , we say the dimension of U is less than the dimension of V , and write dim(U) ≤ dim(V ), if there exists a partition of U into a finite number of definable subsets U1, . . . , Un and definable functions fi : Ui → V , for 1 ≤ i ≤ n, such that the fi’s have finite fibres (i.e. the preimage of any point is of size less than m for some fixed natural number m). By “definable”, we mean what others may call “interpretable”, that is to say, we live in the structure M; we also allow parameters in definitions. However, we use here the notion of dimension for sets definable without parameters in the cartesian product of the considered structure, which will be a field K. As in set theory without choice where we introduce the expression |U | ≤ |V | to mean that we have an injection from U to V without worrying about defining an entity called the “cardinality of U”, we consider the expression “dim(U) ≤ dim(V )” on its own. The important thing is the transitivity: if dim(U) ≤ dim(V ) and dim(V ) ≤ dim(W ), then dim(U) ≤ dim(W ); this implies that the relation of “having the same dimension”, that is, dim(U) ≤ dim(V ) and dim(V ) ≤ dim(U), is an equivalence. So, if the reader would like an object named “the dimension of U”, take the monster model and quotient the set of all of its definable parts by this congruence.