Dirichlet's theorem on diophantine approximation asserts that, for every irrational real number $\alpha$, there are infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1, q > 0$ such that

$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^2}.$$

It can be shown that the numerator $1$ can be replaced with any constant $> 1/\sqrt{5}$ (Hurwitz's theorem).

My question is, if we restrict to real algebraic numbers $\alpha$ with degree $d \geq 3$, can we replace the $1$ in the numerator of Dirichlet's theorem with an arbitrarily large small number? Indeed, we ask:

Let $\alpha$ be an algebraic number of degree $d \geq 3$ and let $c > 0$. Do there exist infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1$ and $q > 0$ such that

$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{c}{q^2}?$$