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Formalizing Norm Extensions and Applications to Number Theory de Frutos-Fernández, María Inés formal mathematics Lean mathlib algebraic number theory p-adic analysis Galois representations p-adic Hodge theory The field ℝ of real numbers is obtained from the rational numbers ℚ by taking the completion with respect to the usual absolute value. We then define the complex numbers ℂ as an algebraic closure of ℝ. The p-adic analogue of the real numbers is the field ℚ_p of p-adic numbers, obtained by completing ℚ with respect to the p-adic norm. In this paper, we formalize in Lean 3 the definition of the p-adic analogue of the complex numbers, which is the field ℂ_p of p-adic complex numbers, a field extension of ℚ_p which is both algebraically closed and complete with respect to the extension of the p-adic norm. More generally, given a field K complete with respect to a nonarchimedean real-valued norm, and an algebraic field extension L/K, we show that there is a unique norm on L extending the given norm on K, with an explicit description. Building on the definition of ℂ_p, we formalize the definition of the Fontaine period ring B_{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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik María Inés de Frutos-Fernández 2023 Is Part Of LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITP.2023.13 urn:nbn:de:0030-drops-183880 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.13 eng https://creativecommons.org/licenses/by/4.0/legalcode