Papers › A Formalization of Complete Discrete Valuation Rings and Local Fields
A Formalization of Complete Discrete Valuation Rings and Local Fields
María Inés de Frutos-Fernández, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio
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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 ℚₚ of p-adic numbers and of the field 𝔽ₚ((X)) of Laurent series over 𝔽ₚ are local fields.
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