Let $K$ be a perfectoid field, if $K$ has positive characterestic then every $a \in K$ has any $p$ power roots i.e $a=x^{p^n}$ have solutions for any positive integer $n$.

However, in characteristic zero case there maybe elements do not admit $p$ power roots. Anyway, are them form a opensubset in $K$? If not, do they at least contain a open subset?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.