An expansion of F̃p

Abstract

Let K be a field of characteristic p. The map τ(X) = Xp − X is an additive endomorphism of K, with kernel Fp. The Galois extensions of K of order p are obtained by adjoining to K solutions to equations of the form Xp − X = a for some a in K. These extensions are called the Artin-Schreier extensions of K and have a cyclic Galois group.The study of Artin-Schreier extensions is very important for studying fields of characteristic p, in particular for studying valued fields of the form K((t)). An attempt at getting quantifier elimination for those fields would necessitate the adjunction to the language of fields of a cross-section for the function τ, i.e. a function σ such that τ ∘ σ is the identity on the image of τ. When K = Fp, such a cross-section is in fact definable in K((t)): it associates to τ(x) the element of {x, x + 1, …, x + p – 1} whose constant term is 0 (see [2]). When K is infinite, such a cross-section is usually not definable.The results presented in this paper originate from a question of L. van den Dries: is there a natural way of defining a cross-section σ for τ on F̃p, and is the theory of (F̃p, σ) decidable? (F̃p is the algebraic closure of Fp.)