Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three

Abstract

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 $\Bbb F_p$ , p being a prime. As the main result, the following theorem will be proved: Let D ∈ Z $D\in \Bbb Z$ and let CD be the set of all cubic polynomials x 3 + a x 2 + b x + c ∈ Z [ x ] $x^3+ax^2+bx+c\in\Bbb Z[x]$ with a discriminant equal to D. If D is square-free and 3 ∤ h ( − 3 D ) $3\nmid h(-3D)$ where h ( − 3 D ) $h(-3D)$ is the class number of Q ( − 3 D ) $\Bbb Q(\sqrt {-3D})$ , then all cubic polynomials in CD have the same type of factorization over any Galois field F p $\Bbb F_p$ where p is a prime, p > 3.