Let $L$ be a finite extension of $\mathbb{Q}_p$, say $[L:\mathbb{Q}_p]=n$. The uniformizer of $\mathbb{Q}_p$ is $p$, let we assume that $\sqrt[n]{p} \notin L$ and consider $K=L(\sqrt[n]{p})$. Is $\pi_L\cdot \sqrt[n]{p}$ the uniformizer of $K$ where $\pi_L$ is the uniformizer of $L$?
2026-03-26 02:56:19.1774493779
A uniformizer of a finite extension of $\mathbb{Q}_p$
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