I encountered a hard problem as below.
Find all the real numbers having the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be written as linear combinations (with rational coefficients) of some roots of unity, where $p,q$ are prime numbers (can be the same) and $a$ is a positive integer with $a>1$ that is not the $q$ th power of an integer.
I tried something below.
Let $r=\sqrt[p]{2021+\sqrt[q]{a}}$, then $r$ can be written as a polynomial of a root of unity $\zeta$ where $\zeta^n=1$. Let $r=g(\zeta)$. Let $h(x)=(g^p(x)-2021)^q-a$, then $h(\zeta)=0$. I quit here. Simply by looking at the degree of $\mathbb{Q}(r)/\mathbb{Q}$ or $\mathbb{Q}(\zeta)/\mathbb{Q}$ will not solve the problem. The biggest problem is we don't know the degree of $g(x)$.
I don't have the correct answer of this problem. How to solve it?