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
Let D ∈ ℤ and let C
D
be the set of all monic cubic polynomials x
3 + ax
2 + bx + c ∈ ℤ[x] with the discriminant equal to D. Along the line of our preceding papers, the following Theorem has been proved: If D is square-free and 3 ∤ h(−3D) where h(−3D) is the class number of
Q
(
−
3
D
)
,
$\mathbb Q(\sqrt {-3D}),$
then all polynomials in C
D
have the same type of factorization over the Galois field 𝔽
p
where p is a prime, p > 3. In this paper, we prove the validity of the above implication also for primes 2 and 3.
Let D ∈ ℤ and let CD be the set of all monic cubic polynomials with integer coefficients having a discriminant equal to D. In this paper, we devise a general method of establishing whether, for a prime p, all polynomials in CD have the same type of factorization over the Galois field $\Bbb F_p$ .
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