An expansion of F̃ p

Let K be a field of characteristic p . The map τ ( X ) = X p − X is an additive endomorphism of K , with kernel F p . The Galois extensions of K of order p are obtained by adjoining to K solutions to equations of the form X p − 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 = F p , 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 F̃ p , and is the theory of (F̃ p , σ ) decidable? (F̃ p is the algebraic closure of F p .)