TITLE: Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.
QUESTION [10 upvotes]: Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$?
Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.

REPLY [3 votes]: I gave this question to a student as a summer project. Thanks to Mercio's answer, we have worked out how to find explicit uniformizers of $\mathbb{Q}_p(\zeta_{p^2},p^{\frac{1}{p}})$ for all odd $p$. We have written a short article on our calculations, which can be found here: http://arxiv.org/abs/1912.01656 Please let us know if you have any suggestions and/or comments for us.