PreprintCC BY 4.0arXiv
Necessary and sufficient condition for reciprocal polynomials to be monogenic
Abstract
Let ℤK denote the ring of integers of the number field K=ℚ(θ), where θ is a root of a monic irreducible polynomial f(x)∈ℤ[x]. We say that f(x) is monogenic if ℤK=ℤ[θ]. A polynomial f(x)∈ℤ[x] is called reciprocal if f(x)=x°(f)f(1/x). In this article, we establish necessary and sufficient conditions for the monogeneity of reciprocal polynomials. As an application, we obtain an alternative and much simpler proof that the maximal real subfields of cyclotomic fields are monogenic.
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BibTeX
@misc{narode2026necessary,
title = {{Necessary and sufficient condition for reciprocal polynomials to be monogenic}},
author = {Anuj Narode},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33766},
eprint = {2609.33766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33766},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Necessary and sufficient condition for reciprocal polynomials to be monogenic
AU - Anuj Narode
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33766
UR - https://arxiv.org/abs/2609.33766
AB - Let ℤ_K denote the ring of integers of the number field K=ℚ(θ), where θ is a root of a monic irreducible polynomial f(x)∈ℤ[x]. We say that f(x) is monogenic if ℤ_K=ℤ[θ]. A polynomial f(x)∈ℤ[x] is called reciprocal if f(x)=x^°(f)f(1/x). In this article, we establish necessary and sufficient conditions for the monogeneity of reciprocal polynomials. As an application, we obtain an alternative and much simpler proof that the maximal real subfields of cyclotomic fields are monogenic.
ER -