  In this paper, we study cyclic codes over the Galois ring ${\rm
GR}({p^2},s)$. The main result is the characterization and enumeration of
Hermitian self-dual cyclic codes of length $p^a$ over ${\rm GR}({p^2},s)$.
Combining with some known results and the standard Discrete Fourier Transform
decomposition, we arrive at the characterization and enumeration of Euclidean
self-dual cyclic codes of any length over ${\rm GR}({p^2},s)$. Some corrections
to results on Euclidean self-dual cyclic codes of even length over
$\mathbb{Z}_4$ in Discrete Appl. Math. 128, (2003), 27 and Des. Codes Cryptogr.
39, (2006), 127 are provided.
