Witt vector

Article summary

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(\mathbb{F}_p) over the finite field of prime order p is isomorphic to \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_1p+a_2p^2 + \cdots to represent an element in \mathbb{Z}_p , an expansion using the Teichmüller character can be considered instead; \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 in \mathbb{F}_p to an element in the solution set of x^{p-1}-1 in \mathbb{Z}_p . That is, the elements in \mathbb{Z}_p can be expanded out in terms of roots of unity instead of as profinite elements in \prod \mathbb{F}_p . We also set \omega(0)=0 , which defines an injective multiplicative map \omega: \mathbb{F}_p \to \mathbb{Z}_p sending elements of \mathbb{F}_p to roots of x^p-x in \mathbb{Z}_p .…

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