On cubic equations over \(p\)-adic fields
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.11124
DOI: 10.1142/S1793042114500201
author: Mansoor Saburov
author: Farruh Mukhamedov
author: B. A. Omirov
published in: International Journal of Number Theory
publication date: 2014-08-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.1743
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D88
Mathematics Subject Classification ID: 12E10
zbMATH DE Number: 6331998
zbMATH Keywords: cubic equations
zbMATH Keywords: \(p\)-adic fields
zbMATH Keywords: solvability criteria
zbMATH Keywords: numbers of solutions
zbMATH Keywords: \(p\)-adic Cardano formula
MaRDI profile type: Publication
arXiv ID: 1204.1743
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title: On cubic equations over \(p\)-adic fields
review text: For a prime number \(p>3\), let \(\mathbb{Q}_p\) denote the field of \(p\)-adic numbers and let \(\mathbb{Z}_p\) denote the ring of integers in \(\mathbb{Q}_p\). The authors provide a criterion for solvability in \(\mathbb{Z}_p^{\times}\), \(\mathbb{Z}_p\) and \(\mathbb{Q}_p\) of any cubic equation over \(\mathbb{Q}_p\). Moreover, the numbers of solutions of them are respectively calculated. Finally, the cubic equations over \(\mathbb{Q}_p\) for which the Cardano method is applicable, are determined.
reviewed by: Manabu Yoshida
description: scientific article; zbMATH DE number 6331998