An archimedean field is an ordered field in which every element is bounded above by a natural number.
So an archimedean field has no infinite elements (and thus no non-zero infinitesimal elements).
For $k$ a non-archimedean field for some non-archimedean absolute value ${\vert -\vert}$ one defines
its ring of integers to be
This is a local ring with maximal ideal
The residue field of $k$ is the quotient
Archimedean fields include
Non-archimedean fields include