{"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/formalizing-norm-extensions-and-applications","title":"Formalizing Norm Extensions and Applications to Number Theory","arxiv_id":"2306.17234","date":"2023-06-29","proceeding":null,"authors":["María Inés de Frutos-Fernández"],"abstract":"Let $K$ be a field complete with respect to a nonarchimedean real-valued norm, and let $L/K$ be an algebraic extension. We show that there is a unique norm on $L$ extending the given norm on $K$, with an explicit description. As an application, we extend the $p$-adic norm on the field $\\mathbb{Q}_p$ of $p$-adic numbers to its algebraic closure $\\mathbb{Q}_p^{\\text{alg}}$, and we define the field $\\mathbb{C}_p$ of $p$-adic complex numbers as the completion of the latter with respect to the $p$-adic norm. Building on the definition of $\\mathbb{C}_p$, we formalize the definition of the Fontaine period ring $B_{\\text{HT}}$ and discuss some applications to the theory of Galois representations and to $p$-adic Hodge theory. The results formalized in this paper are a prerequisite to formalize Local Class Field Theory, which is a fundamental ingredient of the proof of Fermat's Last Theorem.","url_abs":"https://arxiv.org/abs/2306.17234v1","url_pdf":"https://arxiv.org/pdf/2306.17234v1.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":"formalizing-norm-extensions-and-applications","repo_url":"https://github.com/mariainesdff/norm_extensions_journal_submission","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}