Roots of irreducible polynomials in a finite field

Question

If $f$ and $g$ are irreducible polynomials over a finite field $\mathbb F_q$, both of degree $d$, then they both split in $\mathbb F_{q^d}$. One way to represent $\mathbb F_{q^d}$ is to adjoin a root $\alpha$ of $f$. As a basis are the elements $1,\alpha,\ldots,\alpha^{d-1}$. My question is, what are the coordinates of the roots of $g$ with respect to this basis?