On coefficient valuations of Eisenstein polynomials

2003/07/16 by Matthias Kuenzer, M. Kuenzer, Kuenzer, M. +3
Mathematics · #11R18 #11R60 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R18 #msc:11R60

paper · pdf · doi:10.48550/arxiv.math/0307219

arxiv created 2003/07/16 · openalex publication_date 2003/07/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p > 2 be a prime, let n > m > 0. Let pi_n be the norm of zeta_pⁿ - 1 under C_p-1, so that Z_(p)[pi_n] | Z_(p) is a purely ramified extension of discrete valuation rings of degree p^n-1. The minimal polynomial of pi_n over Q(pi_m) 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.

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