Witt vectors

Summary

Fix a prime $p$ and a commutative ring $A$. The ring $W(A)$ of $p$-typical Witt vectors consists of infinite sequences of elements of $A$ with a special addition and multiplication. These operations are built so that certain polynomial functions of the entries, the ghost components, turn them into ordinary addition and multiplication. The construction lifts rings of characteristic $p$ to characteristic $0$: the Witt vectors of the finite field $\F_p$ form the ring $\Z_p$ of p-adic integers. Witt vectors are a basic tool in arithmetic geometry and in the theory of perfect fields of characteristic $p$.