What is "perfectoid"

In mathematics, perfectoid objects occur in the study of problems of " mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p. The notion was introduced by Peter Scholze.

A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements.

Associated to any perfectoid field K there is another K of characteristic p where the multiplication may be defined as

$$K^\flat = \varprojlim_{x \mapsto x^p} K \ .$$

The absolute Galois groups of K and K are isomorphic.