Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$
I want to find a multiplicative inverse of $\sigma$ i .e
$\sigma^{-1}=s+ta\rho+ud\rho^{-1}\in K$ such that
$\sigma\sigma^{-1}=1$
my first idea was to simply multiply these two elements and make it equal to $1$
$(p+qa\rho+rd\rho^{-1})(s+ta\rho+ud\rho^{-1})=1$
$\begin{equation}\begin{split}1&=\\ &=ps+pta\rho+pud\rho^{-1}+sqa\rho+qta^2\rho^2+quad+rsd\rho^{-1}+rtad+rud^2\rho^{-2}\end{split}\end{equation}$
but I have hard time with determining coeffiecients $s,ta,ud$. I get only 4 equations but many more variables. Is there a better way? perhaps with some matrix equation?
Thanks for any help