{"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/sorting-and-labelling-integral-ideals-in-a","title":"Sorting and labelling integral ideals in a number field","arxiv_id":"2005.09491","date":"2020-05-19","proceeding":null,"authors":["John Cremona","Aurel Page","Andrew V. Sutherland"],"abstract":"We define a scheme for labelling and ordering integral ideals of number fields, including prime ideals as a special case. The order we define depends only on the choice of a monic irreducible integral defining polynomial for each field $K$, and we start by defining for each field its unique reduced defining polynomial, after Belabas. We define a total order on the set of prime ideals of $K$ and then extend this to a total order on the set of all nonzero integral ideals of $K$. This order allows us to give a unique label of the form $N.i$, where $N$ is its norm and $i$ is the index of the ideal in the ordered list of all ideals of norm $N$. Our ideal labelling scheme has several nice properties: for a given norm, prime ideals always appear first, and given the factorisation of the norm, the bijection between ideals of norm $N$ and labels is computable in polynomial time. Our motivation for this is to have a well-defined and concise way to sort and label ideals for use in databases such as the LMFDB. We have implemented algorithms which realise this scheme, in Sage, Magma and Pari.","url_abs":"http://arxiv.org/abs/2005.09491v1","url_pdf":"http://arxiv.org/pdf/2005.09491v1.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":"sorting-and-labelling-integral-ideals-in-a","repo_url":"https://github.com/JohnCremona/Sorting","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}