The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of \(p\)-adic fields
type: Item
instance of: scholarly article
title: The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of \(p\)-adic fields
zbMATH Open document ID: 1476.11138
author: Chad Awtrey
author: Alexander Gaura
author: Sebastian Pauli
author: Sandi Rudzinski
author: Ariel Uy
author: Scott Zinzer
published in: Involve
publication date: 2021-06-30T00:00:00Z
review text: This article takes interest to the classification of extensions of \(p\)-adic fields using invariants. The authors introduce a new invariant and show how it is related to other invariants of the extension. Commonly, only the valuation of the discriminant is used as an invariant of extensions of a \(p\)-adic field, since it yields information about the ramified part of the extension. It is natural to ask whether this invariant can be refined by considering not the complete discriminant but possibly some of the digits of its \(p\)-adic expansion. To this end the authors restrict their investigation to discriminants of integral bases of a certain form. They find that the first digit of the \(p\)-adic expansion of the discriminant of generating Eisenstein polynomials is an invariant for most extensions. This allows for a finer classification of totally ramified extensions. The structure of the article is organized in two parts, the first section presents the main results of the paper and the second contains proofs to those results.
reviewed by: Mouad Moutaoukil
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 7364869
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: Eisenstein polynomial
zbMATH Keywords: discriminant
zbMATH Keywords: invariant
MaRDI profile type: Publication
full work available at URL: involve.2020.13.747
OpenAlex ID: W3112571734
cites work: Q5638423
cites work: Ramification polygons, splitting fields, and Galois groups of Eisenstein polynomials
cites work: A database of local fields
cites work: On the computation of all extensions of a \(p\)-adic field of a given degree
DOI: 10.2140/INVOLVE.2020.13.747
description: scientific article; zbMATH DE number 7364869