Publication 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

journal article in Web of Science

Language

en

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

Released

01.06.2017

Publisher

Utilitas Mathematica Publishing

Location

Pages from

99

Pages to

109

Pages count

11

Documents

BibTex


@article{BUT136560,
title="LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE IMAGINARY CASE",
annote="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.",
}