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Let $p\ge 3$ be a prime, let $n\>m\ge 1$. Let ${\pi }_n$ be the norm of ${\zeta }_{p^n}-1$ under $C_{p-1}$, so that ${\mathbb{Z}}_{\left(p\right)}\left[{\pi }_n\right]|{\mathbb{Z}}_{\left(p\right)}$ is a purely ramified extension of discrete valuation rings of degree $p^{n-1}$. The minimal polynomial of ${\pi }_n$ over $\mathbb{Q}\left({\pi }_m\right)$ is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at ${\pi }_m$. The function field analogue, as introduced by Carlitz and Hayes, is studied as well.