Witt vector

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 ${\displaystyle W(\mathbb {F} _{p})}$ over the finite field of prime order p is isomorphic to ${\displaystyle \mathbb {Z} _{p}}$, the ring of p-adic integers. They have a highly non-intuitive structure^[1] 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^[1] behind Witt vectors is that instead of using the standard p-adic expansion

${\displaystyle a=a_{0}+a_{1}p+a_{2}p^{2}+\cdots }$

to represent an element in ${\displaystyle \mathbb {Z} _{p}}$, an expansion using the Teichmüller character can be considered instead;

${\displaystyle \omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}}$,

which is a group morphism sending each element in the solution set of ${\displaystyle x^{p-1}-1}$ in ${\displaystyle \mathbb {F} _{p}}$ to an element in the solution set of ${\displaystyle x^{p-1}-1}$ in ${\displaystyle \mathbb {Z} _{p}}$. That is, the elements in ${\displaystyle \mathbb {Z} _{p}}$ can be expanded out in terms of roots of unity instead of as profinite elements in ${\displaystyle \prod \mathbb {F} _{p}}$. We also set ${\displaystyle \omega (0)=0}$, which defines an injective multiplicative map ${\displaystyle \omega :\mathbb {F} _{p}\to \mathbb {Z} _{p}}$ sending elements of ${\displaystyle \mathbb {F} _{p}}$ to roots of ${\displaystyle x^{p}-x}$ in ${\displaystyle \mathbb {Z} _{p}}$. A p-adic integer can then be expressed as an infinite sum

${\displaystyle a=\omega (a_{0})+\omega (a_{1})p+\omega (a_{2})p^{2}+\cdots }$,

which gives a Witt vector

${\displaystyle (a_{0},a_{1},a_{2},\ldots )\in W(\mathbb {F} _{p})=(\mathbb {F} _{p})^{\mathbb {N} }}$.

Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give ${\displaystyle W(\mathbb {F} _{p})}$ an additive and multiplicative structure such that ${\displaystyle \omega }$ induces a commutative ring homomorphism.