Which numbers are $u$-invariants of perfectoid fields?

Abstract

Let $p\ne 2$ be a prime. We determine the set of numbers that occur as the $u$-invariant of perfectoid subfields of ${\mathbb{C}}_p$. Given any number that is the $u$-invariant of a field of characteristic $p$, we give an explicit construction of a perfectoid field with the same $u$-invariant. In particular, if a number is the $u$-invariant of a field of characteristic $p$, it is also the $u$-invariant of a field of characteristic zero.