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On the monogenity and Galois group of certain classes of polynomials

  • Anuj Jakhar EMAIL logo , Ravi Kalwaniya and Prabhakar Yadav
Published/Copyright: October 15, 2024
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 74 Issue 5

Abstract

We say a monic polynomial g(x) ∈ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, …, θn−1} is a basis for the ring ℤK of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f(x)=xn+c∑i=1n(ax)n−i∈Z[x]andF(x)=xn+c∑i=1nai−1xn−i∈Z[x] be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn, the symmetric group on n letters.

Keywords: Rings of algebraic integers; index of an algebraic integer; power basis; Galois group
MSC 2010: Primary 11R04; Secondary 11R29; 11Y40

The first author is thankful to IIT Madras for NFIG grant RF/22-23/1035/MA/NFIG/009034. The second and third authors are thankful to their respective institutes for financial support.

  1. Communicated by Marco Cantarini

References

[1] Bishnoi, A.—Khanduja, S. K.: A class of trinomials with Galois group Sn, Algebra Colloq. 19(1) (2012), 905–911.10.1142/S1005386712000764Search in Google Scholar

[2] Cohen, H.: A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin-Heidelberg, 1993.10.1007/978-3-662-02945-9Search in Google Scholar

[3] Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Göttingen Abh. 23 (1878), 1–23.Search in Google Scholar

[4] Filaseta, M.—Moy, R.: On the Galois group over ℚ of a truncated binomial expansion, Colloq. Math. 154 (2018), 295–308.10.4064/cm7474-3-2018Search in Google Scholar

[5] Gaál, I.: Diophantine Equations and Power Integral Bases: Theory and Algorithms, 2nd ed., Birk- häuser/Springer, Cham, 2019.10.1007/978-3-030-23865-0Search in Google Scholar

[6] Gaál, I.—Remete, L.: Integral bases and monogenity of composite fields, Exp. Math. 28(2) (2019), 209–222.10.1080/10586458.2017.1382404Search in Google Scholar

[7] Guardia, J.—Montes, J.—Nart, E.: Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012), 361–416.10.1090/S0002-9947-2011-05442-5Search in Google Scholar

[8] Harrington, J.—Jones, L.: Some new polynomial discriminant formulas, Lith. Math. J. 61 (2021), 483–490.10.1007/s10986-021-09540-xSearch in Google Scholar

[9] Hasse, H.: Zahlentheorie, Akademie-Verlag, Berlin, 1963.10.1515/9783112478202Search in Google Scholar

[10] Jakhar, A.—Khanduja, S. K.—Sangwan, N.: On prime divisors of the index of an algebraic integer, J. Number Theory 166 (2016), 47–61.10.1016/j.jnt.2016.02.021Search in Google Scholar

[11] Jakhar, A.—Khanduja, S. K.—Sangwan, N.: Characterisation of primes dividing the index of a trinomial, Int. J. Number Theory 13(10) (2017), 2505–2514.10.1142/S1793042117501391Search in Google Scholar

[12] Jones, L.: A brief note on some infinite families of monogenic polynomials, Bull. Aust. Math. Soc. 100 (2019), 239–244.10.1017/S0004972719000182Search in Google Scholar

[13] Jones, L.: On necessary and sufficient conditions for the monogenity of a certain class of polynomials, Math. Slovaca 72(3) (2022), 591–600.10.1515/ms-2022-0039Search in Google Scholar

[14] Jones, L.: Infinite families of monogenic quadrinomials, quintinomials and sextinomials, Colloq. Math. 169 (2022), 1–10.10.4064/cm8552-4-2021Search in Google Scholar

[15] Jones, L.—Phillips, T.: Infinite families of monogenic trinomials and their Galois groups, Internat. J. Math. 29(5) (2018), Art. ID 1850039.10.1142/S0129167X18500398Search in Google Scholar

[16] Jones, L.—White, D.: Monogenic trinomials with non-squarefree discriminant, Internat. J. Math. 32(13) (2021), Art. ID 2150089.10.1142/S0129167X21500890Search in Google Scholar
Received: 2024-02-15
Accepted: 2024-05-21
Published Online: 2024-10-15
Published in Print: 2024-10-28

© 2024 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2024-0082
Keywords for this article
Rings of algebraic integers; index of an algebraic integer; power basis; Galois group

Articles in the same Issue

  1. On monoids of endomorphisms of a cycle graph
  2. The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
  3. Completely hereditarily atomic OMLS
  4. New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
  5. On the monogenity and Galois group of certain classes of polynomials
  6. Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
  7. Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
  8. Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
  9. Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
  10. The discrete fractional Karamata theorem and its applications
  11. Complex Generalized Stancu-Schurer Operators
  12. New oscillatory criteria for third-order differential equations with mixed argument
  13. Cauchy problem for a loaded hyperbolic equation with the Bessel operator
  14. Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
  15. On the rate of convergence for the q-Durrmeyer polynomials in complex domains
  16. Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
  17. On the relation of Kannan contraction and Banach contraction
  18. Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
  19. Residual Inaccuracy Extropy and its properties
  20. Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
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