Let $K = \mathbb{Q}_5(\zeta_3)$. Consider the basis $\mathcal{B} = (1,\zeta_3)$ of $K$ as a $\mathbb{Q}_5$-space.
Also, let $\zeta_{91}$ be a primitive 91th-root of unity in $\bar{K}$.
Question How can I describe the coeffients of $\min_K(\zeta_{91})$ in terms of the basis $\mathcal{B}$
(i.e. in the form $c_i = a_i+b_i \zeta_3$ where $c_i$ are the coefficients of the minimal polynomial and $a_i,b_i \in \mathbb{Q}_5$)?
Progress:
I know that $K(\zeta_{91})/K$ is unramified and has degree $6$, so the minimal polynomial has degree $6$.
The minimal polynomial of $\zeta_{91}$ over $K$ is (if I recall correctly) $\prod_{j=0}^5 (x-\zeta_{91}^{25^j})$. I thought it might be helpful to look at how this would work over $\mathbb{C}$, so I tried to compute the result in WolframAlpha. However, it did not look very useful to me. I only thought that this result from it might help me:
Is there a better approach to deal with this problem?
Thank you!