In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W ( F p ) {\displaystyle W(\mathbb {F} _{p})} over the finite field of prime order p is isomorphic to Z p {\displaystyle \mathbb {Z} _{p}} , the ring of p-adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers.
The main idea behind Witt vectors is that instead of using the standard p-adic expansion a = a 0 + a 1 p + a 2 p 2 + ⋯ {\displaystyle a=a_{0}+a_{1}p+a_{2}p^{2}+\cdots } to represent an element in Z p {\displaystyle \mathbb {Z} _{p}} , an expansion using the Teichmüller character can be considered instead; ω : F p ∗ → Z p ∗ {\displaystyle \omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}} ,which is a group morphism sending each element in the solution set of x p − 1 − 1 {\displaystyle x^{p-1}-1} in F p {\displaystyle \mathbb {F} _{p}} to an element in the solution set of x p − 1 − 1 {\displaystyle x^{p-1}-1} in Z p {\displaystyle \mathbb {Z} _{p}} . That is, the elements in Z p {\displaystyle \mathbb {Z} _{p}} can be expanded out in terms of roots of unity instead of as profinite elements in ∏ F p {\displaystyle \prod \mathbb {F} _{p}} . We also set ω ( 0 ) = 0 {\displaystyle \omega (0)=0} , which defines an injective multiplicative map ω : F p → Z p {\displaystyle \omega :\mathbb {F} _{p}\to \mathbb {Z} _{p}} sending elements of F p {\displaystyle \mathbb {F} _{p}} to roots of x p − x {\displaystyle x^{p}-x} in Z p {\displaystyle \mathbb {Z} _{p}} . A p-adic integer can then be expressed as an infinite sum a = ω ( a 0 ) + ω ( a 1 ) p + ω ( a 2 ) p 2 + ⋯ {\displaystyle a=\omega (a_{0})+\omega (a_{1})p+\omega (a_{2})p^{2}+\cdots } ,which gives a Witt vector ( a 0 , a 1 , a 2 , … ) ∈ W ( F p ) = ( F p ) N {\displaystyle (a_{0},a_{1},a_{2},\ldots )\in W(\mathbb {F} _{p})=(\mathbb {F} _{p})^{\mathbb {N} }} .Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give W ( F p ) {\displaystyle W(\mathbb {F} _{p})} an additive and multiplicative structure such that ω {\displaystyle \omega } induces a commutative ring homomorphism.
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