TITLE: Polynomial Pell equation
QUESTION [5 upvotes]: Can someone point me in the right direction?

Let $k$ be a field of characteristic $0$ and let $D \in k[x]$ be non-constant. Prove that the ‘polynomial Pell’ equation
  $$f^2 − Dg^2 = 1,\,\,\,\,f, g ∈ k[x]$$
  has no solutions with $g \neq 0$ if $\deg(D) > 2 \deg \operatorname{rad}(D) − 2$.

At first I tried using Mason-Stothers, but I failed to find the answer.

REPLY [4 votes]: It looks like an easy consequence of the Mason–Stothers theorem.
$\deg f^2=\deg Dg^2\Rightarrow2\deg f=\deg D+2\deg g\Rightarrow\deg D=2\deg f-2\deg g$
$2\deg f<\deg\operatorname{rad}(f^2Dg^2)=\deg\operatorname{rad}(fDg)\le\deg f+\deg\operatorname{rad}(D)+\deg g\Rightarrow\deg D\le2\deg\operatorname{rad}(D)-2$