  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.
