Let
Λ
(
f
)
=
K
[
x
]
[
y
;
f
d
d
x
]
\Lambda (f) = K[x][y; f\frac {d}{dx} ]
be an Ore extension of a polynomial algebra
K
[
x
]
K[x]
over a field
K
K
of characteristic zero where
f
∈
K
[
x
]
f\in K[x]
. For a given polynomial
f
f
, the automorphism group of the algebra
Λ
(
f
)
\Lambda (f)
is explicitly described. The polynomial case
Λ
(
0
)
=
K
[
x
,
y
]
\Lambda (0) = K[x,y]
and the case of the Weyl algebra
A
1
=
K
[
x
]
[
y
;
d
d
x
]
A_1= K[x][y; \frac {d}{dx} ]
were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras
Λ
(
f
)
\Lambda (f)
and
Λ
(
g
)
\Lambda (g)
are isomorphic iff
g
(
x
)
=
λ
f
(
α
x
+
β
)
g(x) = \lambda f(\alpha x+\beta )
for some
λ
,
α
∈
K
∖
{
0
}
\lambda , \alpha \in K\backslash \{ 0\}
and
β
∈
K
\beta \in K
. Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras
Λ
(
f
)
\Lambda (f)
. In this paper we complete the picture, i.e. given the polynomial
f
f
we have the explicit description of the automorphism group of
Λ
(
f
)
\Lambda (f)
.
The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest).