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

The absolute Galois groups of K and K^♭ are isomorphic.