Let $k$ be the residue field of characteristic $p>0$ of a complete local ring $R$ with maximal ideal $m$. A $p$-basis for $k$ is a subset $B \subset k$ such that
(i) $k^p(B)=k$
(ii) all monomials of the type $b_1^{e_1}b_2^{e_2}.....b_r^{e_r}$ , $b_i \in B$, $0\leq e_i \leq p-1$ are linearly independent.
Now what I can't understand is how can we show existence of this $p$- basis of any field $k$ with $ch k=p>0$, because as I think if $k$ is perfect field , then it may not have a $p$-basis. Any help will be appreciated, thanks in advanced