The Ax-Kochen-Ershov Theorem

These are the notes of a course for the summer school Model Theory in Bilbao hosted by the Basque Center for Applied Mathematics (BCAM) and the Universidad del Pa\'is Vasco/Euskal Herriko Unibertsitatea in September 2023. The goal of this course is to prove the Ax-Kochen-Ershov (AKE) theorem. This classical result in model theory was proven by Ax and Kochen and independently by Ershov in 1965-1966. The AKE theorem is considered as the starting point of the model theory of valued fields and witnessed numerous refinements and extensions. To a certain measure, motivic integration can be considered as such. The AKE theorem is not only an important result in model theory, it yields a striking application to $p$-adic arithmetics. Artin conjectured that all $p$-adic fields are $C_2$ (every homogeneous polynomial of degree $d$ and in $>d^2$ variable has a non trivial zero). A consequence of the AKE theorem is that the $p$-adics are asymptotically $C_2$. The conjecture of Artin has been disproved by Terjanian in 1966, yielding that the solution given by the AKE theorem is in a sense optimal. The proof presented here is due to Pas but the general strategy stays faithful to the original paper of Ax and Kochen, which consist in the study of the asymptotic first-order theory of the $p$-adics.