On cubic equations over p-adic fields

Farrukh Mukhamedov; Bakhrom Omirov; Mansoor Saburov

doi:10.1142/S1793042114500201

On cubic equations over p-adic fields

Farrukh Mukhamedov
, Bakhrom Omirov
, Mansoor Saburov

International Islamic University Malaysia
National University of Uzbekistan named after Mirzo Ulugbek

Research output: Contribution to journal › Article › peer-review

38 Scopus citations

Abstract

We provide a solvability criterion for a cubic equation in domains Z _p, Z_p, Q_p. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Z_p, Z _p, Q_p are provided. Since _p is a subgroup of _p, we generalize Serre's and Sun's results concerning with cubic equations over the finite field _p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.

Original language	English
Pages (from-to)	1171-1190
Number of pages	20
Journal	International Journal of Number Theory
Volume	10
Issue number	5
DOIs	https://doi.org/10.1142/S1793042114500201
State	Published - Aug 2014
Externally published	Yes

Keywords

Cubic equation
p-adic Cardano formula
p-adic number
solvability criterion

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10.1142/S1793042114500201

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title = "On cubic equations over p-adic fields",

abstract = "We provide a solvability criterion for a cubic equation in domains Z p, Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Zp, Z p, Qp are provided. Since p is a subgroup of p, we generalize Serre's and Sun's results concerning with cubic equations over the finite field p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.",

keywords = "Cubic equation, p-adic Cardano formula, p-adic number, solvability criterion",

author = "Farrukh Mukhamedov and Bakhrom Omirov and Mansoor Saburov",

year = "2014",

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doi = "10.1142/S1793042114500201",

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T1 - On cubic equations over p-adic fields

AU - Mukhamedov, Farrukh

AU - Omirov, Bakhrom

AU - Saburov, Mansoor

PY - 2014/8

Y1 - 2014/8

N2 - We provide a solvability criterion for a cubic equation in domains Z p, Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Zp, Z p, Qp are provided. Since p is a subgroup of p, we generalize Serre's and Sun's results concerning with cubic equations over the finite field p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.

AB - We provide a solvability criterion for a cubic equation in domains Z p, Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Zp, Z p, Qp are provided. Since p is a subgroup of p, we generalize Serre's and Sun's results concerning with cubic equations over the finite field p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.

KW - Cubic equation

KW - p-adic Cardano formula

KW - p-adic number

KW - solvability criterion

UR - https://www.scopus.com/pages/publications/84904542880

U2 - 10.1142/S1793042114500201

DO - 10.1142/S1793042114500201

M3 - 文章

AN - SCOPUS:84904542880

SN - 1793-0421

VL - 10

SP - 1171

EP - 1190

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 5

ER -

On cubic equations over p-adic fields

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Keywords

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