Let $k$ be an infinite field, and consider the affine line $\mathbb{A}_k^1$ over $k$. We know that every isomorphism $\varphi:\mathbb{A}_k^1\longrightarrow\mathbb{A}_k^1$ is of the form $\varphi(x)=ax+b$ where $a,b\in k$ with $a\not=0$. We prove this by describing all possible $k$-algebra isomorphisms $\varphi^*:k[x]\longrightarrow k[x]$.
However, things get complicated when $k$ is a finite field of characteristic $p$. Note that the coordinate ring of $\mathbb{A}_k^1$ is now the quotient $k[x]/I(\mathbb{A}_k^1)$ where $I(\mathbb{A}_k^1)$ is no more the zero ideal. The theory of permutational polynomials, that is, of those where we have that $\varphi$ is at least bijective, seems utterly complicated. Even worse, the so-called Frobenius map $x\mapsto x^p$ is known to be regular and bijective, but it is not an isomorphism of affine algebraic sets since the corresponding $k$-algebra morphism is not surjective.
I'm wondering if we can prove somehow that, even in case of finite fields, linear maps of the form above are the only possible isomorphisms of the affine line.