Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three
Jiří Klaška, Ladislav Skula · Mathematica Slovaca · 2016
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 Fp $\Bbb F_p$ ,pbeing a prime. As the main result, the following theorem will be proved: Let D∈Z $D\in \Bbb Z$ and letCDbe the set of all cubic polynomials x3+ax2+bx+c∈Z[x] $x^3+ax^2+bx+c\in\Bbb Z[x]$ with a discriminant equal toD. IfDis square-free and 3∤h(−3D) $3 mid h(-3D)$ where h(−3D) $h(-3D)$ is the class number of Q(−3D) $\Bbb Q(\sqrt {-3D})$ , then all cubic polynomials inCDhave the same type of factorization over any Galois field Fp $\Bbb F_p$ wherepis a prime,p> 3.