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Communication Dans Un Congrès Année : 2024

A Formalization of Complete Discrete Valuation Rings and Local Fields

Résumé

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued fields. In particular, we prove that the unit ball with respect to a discrete valuation on a field is a discrete valuation ring and, conversely, that the adic valuation on the field of fractions of a discrete valuation ring is discrete. We define finite extensions of valuations and of discrete valuation rings, and prove some global-to-local results. Building on this general theory, we formalize the abstract definition and some fundamental properties of local fields. As an application, we show that finite extensions of the field $\mathbb{Q}_p$ of $p$-adic numbers and of the field $\mathbb{F}_p(\!(X)\!)$ of Laurent series over $\mathbb{F}_p$ are local fields.
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Dates et versions

ujm-04222610 , version 1 (03-10-2023)
ujm-04222610 , version 2 (13-12-2023)
ujm-04222610 , version 3 (07-01-2024)

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María Inés de Frutos-Fernández, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio. A Formalization of Complete Discrete Valuation Rings and Local Fields. 13th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP ’24), Jan 2024, London, United Kingdom. pp.190-204, ⟨10.1145/3636501.3636942⟩. ⟨ujm-04222610v3⟩
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