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

journal article in Web of Science

en

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.

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.

Cubic polynomial, type of factorization, discriminant

15.03.2017

De Gruyter

Slovakia

0139-9918

67

1

71–82

12