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> Let $g\in \mathbb{Q}[x]$ be a polynomial of degree 2022. Prove that there exist infinitely many rational $q\in (0,1)$ such that $\sqrt[5]{g(q)}\notin \mathbb{Q}$.
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I encountered the above problem in the problem solving seminar. I have seen the following somewhat similar problem;

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> Let $f(x)\in \mathbb{Z}[x]$ be a polynomial of degree 3. Prove that there exist infinitely many integer $n$ such that $f(n)$ is not a perfect square.
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However, the methods used in the latter seem to be not applicable to the former. For the second one, we consider $f(x)=ax^3+bx^2+cx+d$ with $a\neq 0$. Then, if $a<0$, it's trivial. Finally, consider the case when $a>0$ which required results from several other problems we proved in the seminar.

At this point, I don't know other approaches. Since the problem is dealing with a polynomial of degree 2022 this time, I bet there should be a clever way. In addition, the instructor told us this one is much simpler.