  In this paper we investigate repeated root cyclic and negacyclic codes of
length $p^rm$ over $\mathbb{F}_{p^s}$ with $(m,p)=1$. In the case $p$ odd, we
give necessary and sufficient conditions on the existence of negacyclic
self-dual codes. When $m=2m'$ with $m'$ odd, we characterize the codes in terms
of their generator polynomials. This provides simple conditions on the
existence of self-dual negacyclic codes, and generalizes the results of Dinh
\cite{dinh}. We also answer an open problem concerning the number of self-dual
cyclic codes given by Jia et al. \cite{jia}.
