  A Hopf Galois structure on a finite field extension $L/K$ is a pair
$(\mathcal{H},\mu)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf
algebra and $\mu$ a Hopf action. In this paper, we present several results on
Hopf Galois structures on odd prime power degree separable field extensions. We
prove that if a separable field extension of odd prime power degree has a Hopf
Galois structure of cyclic type, then it has no structure of noncyclic type. We
determine the number of Hopf Galois structures of cyclic type on a separable
field extension of degree $p^n$, $p$ an odd prime, such that the Galois group
of its normal closure is a semidirect product $C_{p^n}\rtimes C_D$ of the
cyclic group of order $p^n$ and a cyclic group of order $D$, with $D$ prime to
$p$. We characterize the transitive groups of degree $p^3$ which are Galois
groups of the normal closure of a separable field extension having some cyclic
Hopf Galois structure and determine the number of those. We prove that if a
separable field extension of degree $p^3$ has a nonabelian Hopf Galois
structure then it has an abelian structure whose type has the same exponent as
the nonabelian type. We obtain that, for $p>3$, the two abelian noncyclic Hopf
Galois structures do not occur on the same separable extension of degree $p^3$.
We present a table which gives the number of Hopf Galois structures of each
possible type on a separable extension of degree $27$ to illustrate that for
$p=3$, all four noncyclic Hopf Galois structures may occur on the same
extension. Finally, putting together all previous results, we list all possible
sets of Hopf Galois structure types on a separable extension of degree $p^3$,
for $p>3$ a prime.
