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 order ${\displaystyle p}$ is isomorphic to ${\displaystyle \mathbb {Z} _{p}}$, the ring of ${\displaystyle p}$-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 instead of using the standard ${\displaystyle p}$-adic expansion

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

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

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

which sends 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, we expand out elements in ${\displaystyle \mathbb {Z} _{p}}$ in terms of roots of unity instead of as profinite elements in ${\displaystyle \prod \mathbb {F} _{p}}$. We can then express a ${\displaystyle p}$-adic integer as an infinite sum

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

which gives a Witt vector

${\displaystyle (\omega (a_{0}),\omega (a_{1}),\omega (a_{2}),\ldots )}$

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 morphism.