Formalizing Norm Extensions and Applications to Number Theory
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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 ℚ_p of p-adic numbers to its algebraic closure ℚ_p^alg, and we define the field 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 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.
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