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

Jiří Klaška 1  and Ladislav Skula 2
  • 1 Institute of Mathematics Faculty of Mechanical Engineering Brno University of Technology Technická 2 616 69 Brno Czech Republic
  • 2 Department of Mathematics and Statistics Faculty of Science Masaryk University Kotlářská 2 611 37 Brno Czech Republic
Jiří Klaška and Ladislav Skula

Abstract

Let D ∈ ℤ and let CD be the set of all monic cubic polynomials x3 + ax2 + 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(3D), then all polynomials in CD have the same type of factorization over the Galois field 𝔽p where p is a prime, p > 3. In this paper, we prove the validity of the above implication also for primes 2 and 3.

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    Klaška, J.—Skula, L.: Mordell’s equation and the Tribonacci family, Fibonacci Quart. 49 (2011), 310–319.

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    Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials – the real case, to appear in Util. Math.

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    Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials – the imaginary case, to appear in Util. Math.

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    Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials – the case of discriminants divisible by three, Math. Slovaca 66 (2016), 1019–1027.

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    Ward, M.: The characteristic number of a sequences of integers satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), 153–165.

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Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

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