On the factorizations of cubic polynomials with the same discriminant modulo a prime
-
Jiří Klaška
and
Ladislav Skula
Abstract
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
Acknowledgement
The authors will be grateful to any anonymous referee for carefully reading and commenting this paper.
References
[1] Klaška, J.—Skula, L.: Mordell’s equation and the Tribonacci family, Fibonacci Quart. 49(4) (2011), 310-319.Search in Google Scholar
[2] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the real case, Util. Math. 102 (2017), 39-50.Search in Google Scholar
[3] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the imaginary case, Util. Math. 103 (2017), 99-109Search in Google Scholar
[4] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the case of discriminants divisible by three, Math. Slovaca 66(4) (2016), 1019-1027.10.1515/ms-2015-0199Search in Google Scholar
[5] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the case of primes 2 and 3, Math. Slovaca 67(1) (2017), 71-82.10.1515/ms-2016-0248Search in Google Scholar
[6] Kučera, R.: Revealing two cubic non-residues in a quadratic field locally, Math. Slovaca 68(1) (2018), 53-56.10.1515/ms-2017-0079Search in Google Scholar
[7] Stickelberger, L.: Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper, Verhand. I. Internat. Math. Kongress (1897), 182-193.Search in Google Scholar
[8] Voronoï, G.: On Integral Algebraic Numbers Depending on a Root of an Irreducible Equation of the Third Degree, Master’s dissertation (in Russian), 1894.Search in Google Scholar
[9] Voronoï, G.: Sur une propriété du discriminant des fonctions enti`eres, Verhand. III. Internat. Math. Kongress (1905), 186-189.Search in Google Scholar
[10] Ward, M.: The characteristic number of a sequences of integers satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), 153-165.10.1090/S0002-9947-1931-1501582-XSearch in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
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- On the proximity of large primes
- On the factorizations of cubic polynomials with the same discriminant modulo a prime
- Some results on abstract convexity of functions
- Remarks on b-Metric and metric-preserving functions
- Generalization of Ostrowski inequality for convex functions
- On microscopic sets and Fubini Property in all directions
- New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator
- Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations
- Approximate sentinels for diffusion phenomena with pollution
- Remarks on a semilinear system in ℝn motivated by difference equations
- Oscillation tests for difference equations with several non-monotone deviating arguments
- On the regularity of one-sided fractional maximal functions
- A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space
- A note about Volterra operator
- Some reverse and numerical radius inequalities
- A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds
- Sequential decreasing strong size properties
- Approximation of Information Divergences for Statistical Learning with Applications
- Wigner's theorem for an infinite set
- A note on automorphisms of lie ideals in prime rings
Articles in the same Issue
- Prof. RNDr. Beloslav Riečan, DrSc., Dr.h.c. *NOV. 10, 1936 – †AUG. 13, 2018
- On the n × n × n Rubik's Cube
- Congruences involving alternating harmonic sums modulo pαqβ
- On the proximity of large primes
- On the factorizations of cubic polynomials with the same discriminant modulo a prime
- Some results on abstract convexity of functions
- Remarks on b-Metric and metric-preserving functions
- Generalization of Ostrowski inequality for convex functions
- On microscopic sets and Fubini Property in all directions
- New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator
- Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations
- Approximate sentinels for diffusion phenomena with pollution
- Remarks on a semilinear system in ℝn motivated by difference equations
- Oscillation tests for difference equations with several non-monotone deviating arguments
- On the regularity of one-sided fractional maximal functions
- A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space
- A note about Volterra operator
- Some reverse and numerical radius inequalities
- A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds
- Sequential decreasing strong size properties
- Approximation of Information Divergences for Statistical Learning with Applications
- Wigner's theorem for an infinite set
- A note on automorphisms of lie ideals in prime rings
