Isomorphism Problems and Groups of Automorphisms for Ore Extensions \(K[x][y; f\frac{d}{dx} ]\) (Prime Characteristic)

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