Let $L/K$ be a finite separable field extension and let $\theta$ be a primitive element for $L/K$ with minimal polynomial $\mu(t) \equiv \mu_{\theta/K}(t) = \sum_{k=0}^n c_k t^k$. I am trying to compute the powers $\theta^n,\dots,\theta^{2n-2}$ in terms of $1,\theta,\dots,\theta^{n-1}$. In other words, I am trying to compute the structure constants of the basis $1,\theta,\dots,\theta^{n-1}$ in terms of the coefficients of $\mu(t)$, but the expressions quickly become unmanageable. I can hardly imagine that I am the first one to attempt such a calculation, so my question is

Are there perhaps some nice general formulas or manageable expressions/patterns that capture the structure constants of such a basis in terms of the coefficients of $\mu(t)$? Is there some general context where these appear? Or is it hopeless?

Notice that using the dual basis with respect to the trace essentially seems to run into the same issues. I have looked into J.S.Milne's notes on Fields and Galois Theory and on Algebraic Number Theory as well as in Neukirch's Algebraic Number Theory, but in this regard they don't seem to go beyond calculating the discriminant.