TITLE: Is a local field perfect?
QUESTION [6 upvotes]: Is any local field (with finite residue field) perfect (even with nonzero characteristics)?
I know any finite field is perfect. I studied local fields by Neukirch's book.

REPLY [11 votes]: A complete, discretely valued field of characteristic $p > 0$ and finite residue field $\mathbb{F}_q$ must be isomorphic to the Laurent series field $\mathbb{F}_q((t))$.  In the field $\mathbb{F}_q((t))$ the element $t$ is not a $p$th power -- indeed, it is a uniformizer -- so $\mathbb{F}_q((t))$ is not perfect.  
The same argument shows that if $k$ has characteristic $p > 0$, the Laurent series field $k((t))$ is not perfect.  In fact, apart from finite fields and algebraically closed fields, most natural / interesting fields of positive characteristic are not perfect.  E.g. a field of positive characteristic which is infinite and finitely generated over its prime subfield cannot be perfect.