Let $p$ be a prime, $k$ be an algebraically closed field of characteristic $p$. In this paper, we provide the classification of connected Hopf algebras of dimension $p^3$, except the case when the primitive space of the Hopf algebra is two dimensional and abelian. Each isomorphism class is presented by generators $x, y, z$ with relations and Hopf algebra structures. Let $mu$ be the multiplicative group of $(p^2+p-1)$-th roots of unity. When the primitive space is one-dimensional and $p$ is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by $A_{k}^1/mu$.
Let $p$ be a prime. We complete the classification on pointed Hopf algebras of dimension $p^2$ over an algebraically closed field $k$. When $text{char}k eq p$, our result is the same as the well-known result for $text{char}k=0$. When $text{char}k=p$, we obtain 14 types of pointed Hopf algebras of dimension $p^2$, including a unique noncommutative and noncocommutative type.
