symmetric monoidal (∞,1)-category of spectra
A (complete) archimedean valued field is a field equipped with an archimedean absolute value (and complete with respect to it).
A non-archimedean valued field is one that is not, hence one whose norm satisfies the ultrametric triangle inequality.
One of Ostrowski's theorems says that for $k$ a field complete with respect to an absolute value ${\vert - \vert}$ either the absolute value is archimedean valued, in which case $k$ is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.
For $k$ a non-archimedean valued 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 valued fields include
Non-archimedean valued fields include
Last revised on May 28, 2021 at 15:42:26. See the history of this page for a list of all contributions to it.