  Let $\Lambda (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a
polynomial algebra $K[x]$ over an arbitrary field $K$ of characteristic $p>0$
where $f\in K[x]$. For each polynomial $f$, the automorphism group of the
algebras $\Lambda (f)$ is explicitly described. The automorphism group ${\rm
Aut}_K(\L (f))=S\rtimes G_f$ is a semidirect product of two explicit groups
where $G_f$ is the {\em eigengroup} of the polynomial $f$ (the set of all
automorphisms of $K[x]$ such that $f$ is their common eigenvector). For each
polynomial $f$, the eigengroup $G_f$ is explicitly described. It is proven that
every subgroup of ${\rm Aut}_K(K[x])$ is the eigengroup of a polynomial. It is
proven that the Krull and global dimensions of the algebra $\Lambda (f)$ are 2.
The prime, completely prime, primitive and maximal ideals of the algebra
$\Lambda (f)$ are classified.
