{"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/galois-groups-in-a-family-of-dynatomic","title":"Galois groups in a family of dynatomic polynomials","arxiv_id":"1707.02501","date":"2017-07-08","proceeding":null,"authors":["David Krumm"],"abstract":"For every nonconstant polynomial $f\\in\\mathbb Q[x]$, let $\\Phi_{4,f}$ denote the fourth dynatomic polynomial of $f$. We determine here the structure of the Galois group and the degrees of the irreducible factors of $\\Phi_{4,f}$ for every quadratic polynomial $f$. As an application we prove new results related to a uniform boundedness conjecture of Morton and Silverman. In particular we show that if $f$ is a quadratic polynomial, then, for more than $39\\%$ of all primes $p$, $f$ does not have a point of period four in $\\mathbb Q_p$.","url_abs":"http://arxiv.org/abs/1707.02501v3","url_pdf":"http://arxiv.org/pdf/1707.02501v3.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":"galois-groups-in-a-family-of-dynatomic","repo_url":"https://github.com/davidkrumm/galois_dynatomic","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}