{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/first-degree-prime-ideals-of-composite","title":"First-degree prime ideals of composite extensions","arxiv_id":"2111.04660","date":"2021-11-08","proceeding":null,"authors":["Giordano Santilli","Daniele Taufer"],"abstract":"Let $\\mathbb{Q}(\\alpha)$ and $\\mathbb{Q}(\\beta)$ be linearly disjoint number fields and let $\\mathbb{Q}(\\theta)$ be their compositum. We prove that the first-degree prime ideals of $\\mathbb{Z}[\\theta]$ may almost always be constructed in terms of the first-degree prime ideals of $\\mathbb{Z}[\\alpha]$ and $\\mathbb{Z}[\\beta]$, and vice-versa. We also classify the cases in which this correspondence does not hold, by providing explicit counterexamples. We show that for every pair of coprime integers $d,e \\in \\mathbb{Z}$, such a correspondence almost always respects the divisibility of principal ideals of the form $(e+d\\theta)\\mathbb{Z}[\\theta]$, with a few exceptions that we characterize. Finally, we discuss the computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases.","url_abs":"https://arxiv.org/abs/2111.04660v3","url_pdf":"https://arxiv.org/pdf/2111.04660v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"first-degree-prime-ideals-of-composite","repo_url":"https://github.com/dtaufer/first-degree-prime-ideals","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"first-degree-prime-ideals-of-composite","repo_url":"https://github.com/DTaufer/First-degree-prime-ideals/blob/main/MagmaCode.m","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}