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

Abstract

Let D e Z, D > 0 be square-free, 3 D, and 3 | h(-3D) where h(-3D) is the class number of Q(√-3D). We prove that all cubic polynomials f(x) = x3 + ax2 + bx + c 6 Z[x] with a discriminant D have the same type of factorization over any Galois field Fp where p is a prime, p > 3. Moreover, we show that any polynomial f(x) with such a discriminant D has a rational integer root. a complete discussion of the case D = 0 is also included. © 2015 Utilitas Mathematics.