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 over the finite field of order is isomorphic to , the ring of -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 -adic expansion
to represent an element in , we can instead consider an expansion using the Teichmüller character
which sends each element in the solution set of in to an element in the solution set of in . That is, we expand out elements in in terms of roots of unity instead of as profinite elements in . We can then express a -adic integer as an infinite sum
which gives a Witt vector
Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give an additive and multiplicative structure such that induces a commutative ring morphism.
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