This is a problem from Silverman & Tate's Rational Points on Elliptic Curves. The following is the Diophantine Approximation theorem by Thue which is proved in Chapter 5:
Theorem. Let $b$ be a positive integer which is not a perfect cube, and let $\beta=\sqrt[3]{b}$. Let $C$ be a fixed positive constant. Then there are only finitely many pairs of integers $(p,q)$ with $q>0$ which satisfy the inequality $$\left|\frac{p}{q}-\beta\right|\leq \frac{C}{q^3}.$$
The following is an exercise which I am stuck at:
Exercise 5.9. Let $d\geq 3$ be an integer, and let $b$ be an integer that is not a perfect $d$th power. For any constant $C$, prove that there are only finitely many rational numbers $p/q$ satisfying the inequality $$\left|\frac{p}{q}-\sqrt[d]{b}\right|\leq \frac{C}{q^3}.$$
I can prove this statement using the same argument provided in the book to prove the above theorem, IF $d$ is the degree of the minimal polynomial of $\beta$. The proof constructs an auxiliary polynomial, proves its partial derivatives vanish at $(\beta,\beta)$, proves it is small at around $(\beta,\beta)$, and some of its derivative does not vanish.
The proof is too long, I cannot restate it here. So I'm hoping someone familiar with this method could help me with the case when $d$ is not the degree of the minimal polynomial.
Alternatively, is there a way to prove this exercise using Thue's theorem?
Thank you for your help!