A perfectoid field is a complete non-Archimedean field 
with nondiscrete rank-one valuation
and residue characteristic 
such that the Frobenius
map is surjective on 
, where 
is the valuation ring.
Perfectoid fields provide the base fields for perfectoid
spaces. Their tilts exchange mixed characteristic with characteristic 
while retaining closely related valuation-theoretic structure.
Perfectoid Field
See also
Frobenius Map, Non-Archimedean Geometry, Perfectoid Space, Valuation RingExplore with Wolfram|Alpha
References
Scholze, P. "Perfectoid Spaces." Publ. Math. Inst. Hautes Ètudes Sci. 116, 245-313, 2012. https://doi.org/10.1007/s10240-012-0042-x.Cite this as:
Weisstein, Eric W. "Perfectoid Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PerfectoidField.html