Publication detail

# LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE CASE OF DISCRIMINANTS DIVISIBLE BY THREE

KLAŠKA, J. SKULA, L.

Original Title

LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE CASE OF DISCRIMINANTS DIVISIBLE BY THREE

English Title

LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE CASE OF DISCRIMINANTS DIVISIBLE BY THREE

Type

journal article in Web of Science

Language

en

Original Abstract

In this paper we extend our recent results concerning the validity of the law of inertia for the factorization of cubic polynomials over the Galois field $F_p$, p being a prime. As the main result, the following theorem will be proved: Let $D\in Z$ and let $C_D$ be the set of all cubic polynomials $x^3 +ax^2 +bx+c\in Z[x]$ with a discriminant equal to $D$. If $D$ is square-free and $3\nmid h(-3D)$ where $h(-3D)$ is the class number of $Q(\sqrt(-3D))$, then all cubic polynomials in $C_D$ have the same type of factorization over any Galois field $F_p$ where $p$ is a prime, $p > 3$.

English abstract

In this paper we extend our recent results concerning the validity of the law of inertia for the factorization of cubic polynomials over the Galois field $F_p$, p being a prime. As the main result, the following theorem will be proved: Let $D\in Z$ and let $C_D$ be the set of all cubic polynomials $x^3 +ax^2 +bx+c\in Z[x]$ with a discriminant equal to $D$. If $D$ is square-free and $3\nmid h(-3D)$ where $h(-3D)$ is the class number of $Q(\sqrt(-3D))$, then all cubic polynomials in $C_D$ have the same type of factorization over any Galois field $F_p$ where $p$ is a prime, $p > 3$.

Keywords

cubic polynomial, factorization, Galois field

Released

24.11.2016

Publisher

Location

SK

Pages from

1019

Pages to

1027

Pages count

9

Documents

BibTex


@article{BUT129973,
title="LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE CASE OF DISCRIMINANTS DIVISIBLE BY THREE",
annote="In this paper we extend our recent results concerning the validity of the law of inertia for the factorization of cubic polynomials over the Galois field $F_p$, p being a prime. As the main result, the following theorem will be proved: Let $D\in Z$ and let $C_D$ be the set of all cubic polynomials $x^3 +ax^2 +bx+c\in Z[x]$ with a discriminant equal to $D$.  If $D$ is square-free and $3\nmid h(-3D)$  where  $h(-3D)$ is the class number of $Q(\sqrt(-3D))$, then all cubic polynomials in $C_D$ have the same type of factorization over any Galois field $F_p$ where $p$ is a prime, $p > 3$.
",
}