Law of inertia for the factorization of cubic polynomials -The real case

Abstract

Let D ϵ Z and CD := {f(x) = x3 + ax2 + bx + c ϵ Z[x]; Df = D} where Dj is the discriminant of f(x). Assume that D < 0, D is square-free, 3 D, and 3 h(-3D) where h(-ZD) is the class number of Q(√-3D). We prove that all polynomials in Co have the same type of factorization over any Galois field Fp, p being a prime, p > 3.