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

Abstract

Let D ∈ ℤ and let C D be the set of all monic cubic polynomials x 3 + ax 2 + bx + c ∈ 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 ∤ h(−3D) where h(−3D) is the class number of Q ( − 3 D ) , $\mathbb Q(\sqrt {-3D}),$ 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.

On cubic polynomials with a given discriminant

“…Creating tables of these representatives and analyzing them can lead to the discovery of new interesting facts. These tables can also be useful for a better understanding of the law of inertia for the factorization of cubic polynomials, which has been studied in [11][12][13][14][15][16].…”