Let $f \in \mathbb{Q}[x]$ of degreee $d$ be irreducible, with roots $\alpha_1,\ldots, \alpha_d$. One particular basis for the field extension of $\mathbb{Q}$ obtained by adjoining the roots of $f$ is given by $\{\alpha_j\}_{j=1}^d$. For which $k$ does the set $\{\alpha_j^k\}_{j=1}^d$ also provide a basis?
If this is in general a difficult question, could I at least show that I obtain a basis for infinitely many $k$?
Example: Consider the polynomial $x^2-2$. The $k$th powers of the roots $\{\pm \sqrt{2}\}$ provide a basis for my field extension precisely when $k$ is odd.
Edit: As Gerry Myerson points out, and as my example indicates, I am actually more interested in when the $k$th powers of the roots generate the extension over $\mathbb{Q}$. Calling this a $\mathbb{Q}$-basis was erroneous.