Separably Closed Fields Considered as Modules

P. Dellunde, F. Delon & F. Point

We present some results on the theory of modules of separably closed fields ([2]). Let SCF_{p,e} be the first-order theory of separably closed fields of characteristic p and imperfection degree e ∈ ω ∪ {∞} in the language of fields ([3]). We consider the models of SCF_{p,e} in a weaker language. We fix a model K of SCF_{p,e} and a p-basis B of K. We may consider K as a module over the skew polynomial ring R = 𝔽_{p}(B)[t; α], where α is the Frobenius map (see for instance [1]). We extend the ordinary language of modules with the analog of the p-component functions and we prove that the theory of separably closed fields considered as R-modules, is complete and admits quantifier elimination.

We prove also that our theory (though stable) is not superstable, we give a description of the types and we study the Ziegler space of indecomposables ([4]).

[1] Cohn, P. M., Skew fields, Encyclopedia of mathematics and its applications, edited by G.-C. Rota, volume 57, Cambridge University Press, 1995.

[2] Dellunde, P., Delon, F., Point, F., Theory of modules of separably closed fields, Preprint, 1999.

[3] Ershov Y., Fields with a solvable theory, Doklady 174, 1967, pp. 19-20; English translation in Soviet Mathematics 8, 1967, pp. 575-576.

[4] Ziegler M., Model Theory of modules, Annals of Pure and Applied Logic 26, 1984, pp. 149-213.

Addresses P. Dellunde: Àrea de Lògica, Universitat Autònoma de Barcelona 08193-Bellaterra, Barcelona, Spain

E-mail: Pilar.Dellunde@uab.es

F. Delon: Équipe de Logique Mathématique, UFR de Mathématiques, Case 7012, Université Paris 7-Denis Diderot, 2, Place Jussieu, 75251 Paris, France

E-mail: delon@logique.jussieu.fr

F. Point: Institut de Mathématique et Informatique, Université de Mons-Hainaut, Le Pentagone, 6, avenue du Champ de Mars, B-7000 Mons, Belgium

E-mail: point@logique.jussieu.fr