Formalizing the Ring of Witt Vectors

10/06/2020

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by   Johan Commelin, et al.

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The ring of Witt vectors 𝕎 R over a base ring R is an important tool in algebraic number theory and lies at the foundations of modern p-adic Hodge theory. 𝕎 R has the interesting property that it constructs a ring of characteristic 0 out of a ring of characteristic p > 1, and it can be used more specifically to construct from a finite field containing ℤ/pℤ the corresponding unramified field extension of the p-adic numbers ℚ_p (which is unique up to isomorphism). We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime p, the ring of Witt vectors over ℤ/pℤ is isomorphic to the ring of p-adic integers ℤ_p. In the process we develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature. Our proofs resemble the informal arguments while being fully rigorous.