On necessary and sufficient conditions for the monogeneity of a certain class of polynomials
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Lenny Jones
Abstract
Let f(x) ∈ ℤ[x] be monic and irreducible over ℚ, with deg(f) = n. Let K = ℚ(θ), where f(θ) = 0, and let ℤK denote the ring of integers of K. We say f(x) is monogenic if {1, θ, θ2, …, θn−1} is a basis for ℤK. Otherwise, f(x) is called non-monogenic. In this article, we give necessary and sufficient conditions for a certain class of polynomials to be monogenic. Using these conditions allows us to generate infinite families of non-monogenic polynomials. In particular, for quadrinomials our results show that there exist infinitely many primes p ≥ 3, and integers t ≥ 1 coprime to p, such that f(x) = xp − 2ptxp−1 + p2t2xp−2 + 1 is non-monogenic. Finally, we illustrate this situation with an explicit example.
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(Communicated by István Gaál)
Acknowledgement
The author thanks the anonymous referees for the helpful suggestions.
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© 2022 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- The poset of morphism-extension classes of countable graphs
- Characterization of monadic BL-algebras by state operators
- Comments on efficient batch verification test for digital signatures based on elliptic curves
- On necessary and sufficient conditions for the monogeneity of a certain class of polynomials
- An exponential Diophantine equation involving the sum or difference of powers of two Pell numbers
- Some results on certain types of Putcha semigroups
- Inner functions in QK spaces and multipliers
- Certain subclasses of meromorphic multivalent q-starlike and q-convex functions
- Algebraic dependences of meromorphic mappings into a projective space sharing few hyperplanes
- Unbounded oscillation criteria for fourth order neutral differential equations of non-canonical type
- Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group
- Intuitionistic fuzzy Tribonacci I-convergent sequence spaces
- Lipschitz class functions and their general Fourier coefficients
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Spectra and fine spectra of the generalized upper difference operator with triple repetition
- A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere
- An extended type I half-logistic family of distributions: Properties, applications and different method of estimations
- On the Unit-Chen distribution with associated quantile regression and applications
- A note on Lévy subordinators in cones of fuzzy sets in Banach spaces
- Uniformly asymptotic normality of the weighted estimator in nonparametric regression model with φ-mixing errors
- Upper bound for variance of finite mixtures of power exponential distributions
- The dimension Dind of finite topological T0-spaces
Articles in the same Issue
- Regular Papers
- The poset of morphism-extension classes of countable graphs
- Characterization of monadic BL-algebras by state operators
- Comments on efficient batch verification test for digital signatures based on elliptic curves
- On necessary and sufficient conditions for the monogeneity of a certain class of polynomials
- An exponential Diophantine equation involving the sum or difference of powers of two Pell numbers
- Some results on certain types of Putcha semigroups
- Inner functions in QK spaces and multipliers
- Certain subclasses of meromorphic multivalent q-starlike and q-convex functions
- Algebraic dependences of meromorphic mappings into a projective space sharing few hyperplanes
- Unbounded oscillation criteria for fourth order neutral differential equations of non-canonical type
- Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group
- Intuitionistic fuzzy Tribonacci I-convergent sequence spaces
- Lipschitz class functions and their general Fourier coefficients
-
Spectra and fine spectra of the generalized upper difference operator with triple repetition
- A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere
- An extended type I half-logistic family of distributions: Properties, applications and different method of estimations
- On the Unit-Chen distribution with associated quantile regression and applications
- A note on Lévy subordinators in cones of fuzzy sets in Banach spaces
- Uniformly asymptotic normality of the weighted estimator in nonparametric regression model with φ-mixing errors
- Upper bound for variance of finite mixtures of power exponential distributions
- The dimension Dind of finite topological T0-spaces
