Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3

Rok, strany: 2017, 71 - 82

Let $D\in \Bbb Z$ and let $C_D$ be the set of all monic cubic polynomials $x³+ax²+bx+c\in \Bbb Z[x]$ 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 $3\nmid h(-3D)$ where $h(-3D)$ is the class number of $\Bbb Q(\sqrt {-3D})$, then all polynomials in $C_D$ have the same type of factorization over the Galois field $\Bbb F_p$ where $p$ is a prime, $p>3$. In this paper, we prove the validity of the above implication also for primes $2$ and $3$.

ISO 690:

Klaška, J., Skula, L. 2017. Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3. In Mathematica Slovaca, vol. 67, no.1, pp. 71-82. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0248

APA:

Klaška, J., Skula, L. (2017). Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3. Mathematica Slovaca, 67(1), 71-82. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0248