---
tag: 6N3X2QX4W9
kind: concept
name: Witt vectors
msc: [13F35]
sources:
  - url: https://en.wikipedia.org/wiki/Witt_vector
same_as:
  wikidata: Q1756171
  wikipedia: Witt vector
---

## Summary

Fix a prime $p$ and a [[commutative ring|0310REJJY8]] $A$.
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|0ZXZ82FTX2]] $\F_p$ form the ring $\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].
Witt vectors are a basic tool in arithmetic geometry and in the theory of [[perfect fields|VWBVKT794W]] of characteristic $p$.

## Explanations

### L1 human

A $p$-adic integer can be written as a digit expansion $a_0 + a_1 p + a_2 p^2 + \cdots$ with digits in $\set{0, \ldots, p-1}$.
Adding two such expansions needs carries, and the carry rule does not come from the arithmetic of the digits modulo $p$ alone.
Witt vectors make the carry rule into universal polynomials, so the same construction works with "digits" taken from any ring.
For example, in $W(\F_2)$ the sum of $(1, 0, 0, \ldots)$ with itself is $(0, 1, 0, \ldots)$, just as $1 + 1 = 0 + 1 \cdot 2$ in base $2$.