Suppose we are given an element $\alpha = a+b\sqrt{d}$ in the quadratic number field $\mathbb{Q}(\sqrt{d})$, where $a$, $b$ and $d$ are all rationals and $d$ does not have a rational square root. I want to test whether $\alpha = \beta^3$, for some $\beta \in \mathbb{Q}(\sqrt{d})$.
Using the technique mentioned in this post, one can reduce the above problem to testing whether a suitable cubic equation over $\mathbb{Q}[x]$ has a rational root or not. One can solve that by factoring the cubic polynomial over rational field using the LLL algorithm. But, I do not wish to invoke that big machinery and was wondering if there is an easier and a direct way to decide whether an element has a cubic root in $\mathbb{Q}(\sqrt{d})$, without reducing it to the corresponding cubic equation.
