Detail publikace
LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS – THE IMAGINARY CASE
KLAŠKA, J. SKULA, L.
Anglický název
LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE IMAGINARY CASE
Typ
článek v časopise ve Web of Science, Jimp
Jazyk
en
Originální abstrakt
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.
Anglický abstrakt
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.
Klíčová slova anglicky
cubic polynomial, factorization, Galois field
Vydáno
01.06.2017
Nakladatel
Utilitas Mathematica Publishing
Místo
Canada
ISSN
0315-3681
Ročník
103
Číslo
2
Strany od–do
99–109
Počet stran
11
BIBTEX
@article{BUT136560,
author="Jiří {Klaška} and Ladislav {Skula},
title="LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS – THE IMAGINARY CASE",
year="2017",
volume="103",
number="2",
month="June",
pages="99--109",
publisher="Utilitas Mathematica Publishing",
address="Canada",
issn="0315-3681"
}