Publication result detail

LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE IMAGINARY CASE

KLAŠKA, J.; SKULA, L.

Original Title

LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE IMAGINARY CASE

English Title

LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE IMAGINARY CASE

Type

WoS Article

Original Abstract

Let $D\in \Bbb Z$, $D > 0$ be square-free, $3\nmid D$, and $3 \nmid h(-3D)$ where $h(-3D)$ is the class number of $\Bbb Q(\sqrt(-3D))$. We prove that all cubic polynomials $f(x) = x^3+ax^2+bx+c\in \Bbb Z[x]$ with a discriminant $D$ have the same type of factorization over any Galois field $\Bbb F_p$ where $p$ is a prime, $p > 3$. Moreover, we show that any polynomial $f(x)$ with such a discriminant $D$ has a rational integer root. A complete discussion of the case $D = 0$ is also included.

English abstract

Let $D\in \Bbb Z$, $D > 0$ be square-free, $3\nmid D$, and $3 \nmid h(-3D)$ where $h(-3D)$ is the class number of $\Bbb Q(\sqrt(-3D))$. We prove that all cubic polynomials $f(x) = x^3+ax^2+bx+c\in \Bbb Z[x]$ with a discriminant $D$ have the same type of factorization over any Galois field $\Bbb F_p$ where $p$ is a prime, $p > 3$. Moreover, we show that any polynomial $f(x)$ with such a discriminant $D$ has a rational integer root. A complete discussion of the case $D = 0$ is also included.

Keywords

cubic polynomial, factorization, Galois field

Key words in English

cubic polynomial, factorization, Galois field

Authors

KLAŠKA, J.; SKULA, L.

RIV year

2018

Released

01.06.2017

Publisher

Utilitas Mathematica Publishing

Location

Canada

ISBN

0315-3681

Periodical

UTILITAS MATHEMATICA

Volume

103

Number

2

State

Canada

Pages from

99

Pages to

109

Pages count

11

URL

https://www.degruyter.com/document/doi/10.1515/ms-2016-0248/html

Full text in the Digital Library

http://hdl.handle.net/

BibTex

@article{BUT136560,
  author="Jiří {Klaška} and Ladislav {Skula}",
  title="LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE IMAGINARY CASE",
  journal="UTILITAS MATHEMATICA",
  year="2017",
  volume="103",
  number="2",
  pages="99--109",
  issn="0315-3681",
  url="https://www.degruyter.com/document/doi/10.1515/ms-2016-0248/html"
}