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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 fieldF   p    $\Bbb F_p$   ,  p  being a prime. As the main result, the following theorem will be proved: LetD  ∈   Z    $D\in \Bbb Z$    and let  C D   be the set of all cubic polynomialsx  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 and3  ∤  h  (  −  3  D  )   $3\nmid h(-3D)$    whereh  (  −  3  D  )   $h(-3D)$    is the class number ofQ   (   −  3  D   )   $\Bbb Q(\sqrt {-3D})$   , then all cubic polynomials in  C D   have the same type of factorization over any Galois fieldF   p    $\Bbb F_p$    where  p  is a prime,  p  &gt; 3.

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