On coefficient valuations of Eisenstein polynomials

Abstract

Let p ≥ 3 be a prime, let n > m ≥ 1. Let π n be the norm ofis a purely ramified extension of discrete valuation rings of degree p n−1 . The minimal polynomial of π n over Q(π m ) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at π m . The function field analogue, as introduced by Carlitz and Hayes, is studied as well.