Automorphisms of $\mathbb{Z}[x]$ (verification) Consider the ring $\mathbb{Z}[x]$. If $f$ is an automorphism of this ring, then since $f(1)=1$ so $f$ fixes $\mathbb{Z}$. To determine $f(x)$, since $0$ is a root of this polynomial with multiplicity $1$ we have $f(x)=ax$ for some $a\in\mathbb{Z}$. Applying evaluation map, we get $a=1$. So the only automorphism of $\mathbb{Z}[x]$ is identity.
There could have been attempts to prove that the only automorphisms of the ring under consideration is identity; but my question is whether above proof is correct?
Q. Is this correct way to prove the assertion that $\mathbb{Z}[x]$ only one ring automorphism? (my assertion may be wrong; I don't know in general what it should be.)

Hint: the map
$$
a_0+a_1x+\dots+a_nx^n\mapsto
a_0+a_1(x+1)+\dots+a_n(x+1)^n
$$
is clearly an automorphism of $\mathbb{Z}[x]$
