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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 {\displaystyle W} over the finite field of order p is the ring of p {\displaystyle 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 instead of using the standard p {\displaystyle 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}} , we can instead consider an expansion using the Teichmuller character
ω : F p ∗ → Z p ∗ {\displaystyle \omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}}