Monogenic Fields from Polynomial Compositions with Applications
Abstract
A number field K is called \emph{monogenic} if its ring of integers ℤK can be expressed as a simple ring extension ℤ[α] for some α∈ ℤK. A monic irreducible polynomial f(x)∈ℤ[x] is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for [ℤKi:ℤ[αi]]=1, where Ki=ℚ(αi) and αi is a root of the composed polynomial fi(xk+b) for i=1,2. Here, f1(x)=xn+c∑j=1n(ax)n−j∈ℤ[x] and f2(x)=xn+c∑j=1naj−1xn−j∈ℤ[x] are irreducible polynomials of degree n≥ 3. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.
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BibTeX
@misc{jakhar2026monogenic,
title = {{Monogenic Fields from Polynomial Compositions with Applications}},
author = {Anuj Jakhar and Ravi Kalwaniya and Prabhakar Yadav},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.00949},
eprint = {2605.00949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00949},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monogenic Fields from Polynomial Compositions with Applications
AU - Anuj Jakhar
AU - Ravi Kalwaniya
AU - Prabhakar Yadav
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2605.00949
UR - https://arxiv.org/abs/2605.00949
AB - A number field K is called \emph{monogenic} if its ring of integers ℤ_K can be expressed as a simple ring extension ℤ[α] for some α∈ ℤ_K. A monic irreducible polynomial f(x)∈ℤ[x] is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for [ℤ_K_i:ℤ[α_i]]=1, where K_i=ℚ(α_i) and α_i is a root of the composed polynomial f_i(x^k+b) for i=1,2. Here, f_1(x)=x^n+c∑_j=1^n(ax)^n−j∈ℤ[x] and f_2(x)=x^n+c∑_j=1^na^j−1x^n−j∈ℤ[x] are irreducible polynomials of degree n≥ 3. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.
ER -