$α$-monogeneity of pure number fields: criterion and density

[2510.20232] $α$-monogeneity of pure number fields: criterion and density Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2510.20232 (math) [Submitted on 23 Oct 2025 ( v1 ), last revised 25 Aug 2026 (this version, v3)] Title: $α$-monogeneity of pure number fields: criterion and density Authors: Khai-Hoan Nguyen-Dang , Nguyen Thai Hung View PDF HTML (experimental) Abstract: Let $n\ge 2$, let $m\in\mathbb Z\setminus\{0\}$, and let $K=\mathbb Q(\alpha)$, where $\alpha^n=m$ and $X^n-m$ is irreducible over $\mathbb Q$. We study when the natural order $\mathbb Z[\alpha]$ is the full ring of integers $\mathcal O_K$. For the pure family $X^n-m$, we give a short proof, using only Dedekind's index criterion, of the equivalence $\mathcal O_K = \mathbb Z[\alpha]$ iff $m$ is square-free and $\nu\_p(m^p-m)=1$ for every prime $p\mid n$. Equivalently, the prime support of $[\mathcal O_K:\mathbb Z[\alpha]]$ is $$\{p:\nu_p(m)\ge 2\}\cup \{p\mid n:\nu_p(m^p-m)\ge 2\}.$$ We then compute the natural density of the corresponding parameters in the one-parameter family $X^n-m$: $$\delta_n=\frac{6}{\pi^2}\prod_{p\mid n}\frac{p}{p+1}.$$ We also give an arithmetic-progression refinement, a density-theoretic independence statement for the local obstruction sets at primes dividing $n$, and discriminant-ordered counts of the corresponding fields. Comments: 13 pages, accepted version Subjects: Number Theory (math.NT) Cite as: arXiv:2510.20232 [math.NT] (or arXiv:2510.20232v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2510.20232 Focus to learn more arXiv-issued DOI via DataCite Journal reference: International Journal of Number Theory, 2026 Related DOI : https://doi.org/10.1142/S1793042126501228 Focus to learn more DOI(s) linking to related resources Submission history From: Khai-Hoan Nguyen-Dang [ view email ] [v1] Thu, 23 Oct 2025 05:27:33 UTC (15 KB) [v2] Thu, 8 Jan 2026 03:08:19 UTC (15 KB) [v3] Tue, 25 Aug 2026 03:44:51 UTC (15 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-10 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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