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RAMIFICATION POLYGONS, SPLITTING FIELDS, AND GALOIS GROUPS OF EISENSTEIN POLYNOMIALS

CHRISTIAN GREVE

Florastraße 50, 40217 Düsseldorf, Germany

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and

SEBASTIAN PAULI

Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27402, USA

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https://doi.org/10.1142/S1793042112500832Cited by:7 (Source: Crossref)

Abstract

Let ϕ(x) be an Eisenstein polynomial of degree n over a local field and let α be a root of ϕ(x). The Newton polygon of ρ(x) = ϕ(αx+α)/αⁿ is called the ramification polygon of ϕ(x). We attach an additional invariant, the segmental inertia degree, to each segment of the ramification polygon and use the slopes of the segments and their segmental inertia degrees to describe the splitting field of ϕ(x). Furthermore we present a method for determining the Galois group of ϕ(x) when the ramification polygon consists of one segment.

Keywords:

AMSC: 11S15, 11S20

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