LAW OF INERTIA FOR THE FACTORIZATION OF CUBIC POLYNOMIALS - THE CASE OF DISCRIMINANTS DIVISIBLE BY THREE

článek v časopise ve Web of Science, Jimp

en

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 $F_p$, p being a prime. As the main result, the following theorem will be proved: Let $D\in Z$ and let $C_D$ be the set of all cubic polynomials $x^3 +ax^2 +bx+c\in Z[x]$ with a discriminant equal to $D$. If $D$ is square-free and $3\nmid h(-3D)$ where $h(-3D)$ is the class number of $Q(\sqrt(-3D))$, then all cubic polynomials in $C_D$ have the same type of factorization over any Galois field $F_p$ where $p$ is a prime, $p > 3$.

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 $F_p$, p being a prime. As the main result, the following theorem will be proved: Let $D\in Z$ and let $C_D$ be the set of all cubic polynomials $x^3 +ax^2 +bx+c\in Z[x]$ with a discriminant equal to $D$. If $D$ is square-free and $3\nmid h(-3D)$ where $h(-3D)$ is the class number of $Q(\sqrt(-3D))$, then all cubic polynomials in $C_D$ have the same type of factorization over any Galois field $F_p$ where $p$ is a prime, $p > 3$.

cubic polynomial, factorization, Galois field

24.11.2016

Slovenská akademie věd

SK

0139-9918

66

4

1019–1027

9