Publication detail

# Law of inertia for the factorization of cubic polynomials - the case of primes 2 and 3

KLAŠKA, J. SKULA, L.

Original Title

Law of inertia for the factorization of cubic polynomials - the case of primes 2 and 3

English Title

Law of inertia for the factorization of cubic polynomials - the case of primes 2 and 3

Type

journal article in Web of Science

Language

en

Original Abstract

Let $D \in \mathbb Z$ and let $C_D$ be the set of all monic cubic polynomials $x^3+ax^2+bx+c\in \mathbb Z[x]$ with the discriminant equal to $D$. Along the line of our preceding papers, the following Theorem has been proved: If $D$ is square-free and $3 \nmit h(-3D)$ where $h(-3D)$ is the class number of $\mathbbQ( \sqrt(-3D)$, then all polynomials in $C_D$ have the same type of factorization over the Galois field $F_p$ where $p$ is a prime, $p > 3$. In this paper, we prove the validity of the above implication also for primes 2 and 3.

English abstract

Let $D \in \mathbb Z$ and let $C_D$ be the set of all monic cubic polynomials $x^3+ax^2+bx+c\in \mathbb Z[x]$ with the discriminant equal to $D$. Along the line of our preceding papers, the following Theorem has been proved: If $D$ is square-free and $3 \nmit h(-3D)$ where $h(-3D)$ is the class number of $\mathbbQ( \sqrt(-3D)$, then all polynomials in $C_D$ have the same type of factorization over the Galois field $F_p$ where $p$ is a prime, $p > 3$. In this paper, we prove the validity of the above implication also for primes 2 and 3.

Keywords

Cubic polynomial, type of factorization, discriminant

Released

15.03.2017

Publisher

De Gruyter

Location

Slovakia

Pages from

71

Pages to

82

Pages count

12

Documents

BibTex

```
@article{BUT134703,
author="Jiří {Klaška} and Ladislav {Skula}",
title="Law of inertia for the factorization of cubic polynomials - the case of primes 2 and 3",
annote="Let $D \in \mathbb Z$ and let $C_D$ be the set of all monic cubic polynomials $x^3+ax^2+bx+c\in \mathbb Z[x]$ with the discriminant equal to $D$. Along the line of our preceding papers, the following Theorem has
been proved: If $D$ is square-free and $3 \nmit h(-3D)$ where $h(-3D)$ is the class number of $\mathbbQ(
\sqrt(-3D)$, then all polynomials in $C_D$ have the same type of factorization over the Galois field $F_p$ where $p$ is a prime,
$p > 3$. In this paper, we prove the validity of the above implication also for primes 2 and 3.",
address="De Gruyter",
chapter="134703",
doi="10.1515/ms-2016-0248",
howpublished="print",
institution="De Gruyter",
number="1",
volume="67",
year="2017",
month="march",
pages="71--82",
publisher="De Gruyter",
type="journal article in Web of Science"
}
```