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Let $\L (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomialalgebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For agiven polynomial $f$, the automorphism group of the algebra $\L (f) $ isexplicitly described. The polynomial case $\L (0) = K[x,y]$ and the case of theWeyl algebra $A_1= K[x][y; \frac{d}{dx} ]$ were done done by Jung (1942) andvan der Kulk (1953), and Dixmier (1968), respectively. In 1997, Alev and Dumasproved that the algebras $\L (f)$ and $\L (g)$ are isomorphic iff $g(x) = \lf(\alpha x+\beta )$ for some $\l, \alpha \in K\backslash \{ 0\}$ and $\beta\inK$. In 2015, Benkart, Lopes and Ondrus gave a complete description of the setof automorphism groups of algebras $\L(f)$. In this paper we complete thepicture, i.e. {\em given} the polynomial $f$ we have the explicit descriptionof the automorphism group of $\L (f)$.

Isomorphism problems and groups of automorphisms for Ore extensions $K[x][y; \delta ]$ (zero characteristic)