# Ring of Witt vectors

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## Definition

The term ring of Witt vectors or Witt ring is a commutative unital ring defined in the context of a prime number $p$ and a commutative unital ring $R$. There are two version of this:

• The ring of Witt vectors, by default, refers to infinite sequences, i.e., Witt vectors of infinite length (the detailed definition is given below).
• The ring of Witt vectors of length $d$ refers to the quotient of this ring obtained by simply looking at the first $d$ coordinates of the vector. The operations are defined in a manner that the first $d$ coordinates of the sum and product depend only on the first $d$ coordinates of the vectors being added or multiplied. Thus, it is legitimate to pass to quotients in this manner.

There is a related notion called the ring of universal Witt vectors that makes sense in the context of any commutative unital ring $R$ and does not require specification of any prime number.

### Definition of the ring of Witt vectors of infinite length

We first define the Witt polynomials as the following polynomials in $\mathbb{Z}[X_0,X_1,X_2,\dots,]$ (and hence interpretable in $R[X_0,X_1,X_2,\dots,]$):

$W_n(X) := \sum_{i=0}^n p^iX_i^{p^{n-i}}$

The first few Witt polynomials are given below:

$n$ $W_n$
0 $X_0$
1 $X_0^p + pX_1$
2 $X_0^{p^2} + pX_1^p + p^2X_2$
3 $X_0^{p^3} + pX_1^{p^2} + p^2X_2^p + p^3X_3$

We now consider the set $(X_0,X_1,X_2,\dots,X_n,\dots), X_i \in R$ of all sequences over $R$. We call the elements of this set Witt vectors, and we define the addition and multiplication as follows:

Addition and multiplication are defined in the unique manner so as to make the set of Witt vectors a commutative unital ring with the following two properties: (a) each Witt polynomial is a ring homomorphism from that ring to $R$, and (b) both addition and multiplication are given by polynomials with integer coefficients that depend on $p$ but not on $R$.

Explicitly, the sum and product of the Witt vectors $(X_0,X_1,X_2,\dots,)$ and $(Y_0,Y_1,Y_2,\dots,)$ are given by the following formulas:

$n$ $n^{th}$ coordinate of sum $n^{th}$ coordinate of product
0 $X_0 + Y_0$ $X_0Y_0$
1 $X_1 + Y_1 - \sum_{i=1}^{p-1} \frac{(p-1)!}{i!(p-i)!} X_0^iY_0^{p-i}$, which can be thought of as $X_1 + Y_1 + \frac{(X_0^p + Y_0^p) - (X_0 + Y_0)^p}{p}$ (although the latter notation does not directly make sense unless $p$ is invertible in $R$). $X_0^pY_1 + Y_0^pX_1 + pX_1Y_1$

Some elaboration on why these choices work is provided below:

$n$ $W_n(X)$ $W_n(Y)$ $W_n(X) + W_n(Y)$, which equals $W_n$ of the Witt sum $W_n(X)W_n(Y)$, which equals $W_n$ of the Witt product Comment
0 $X_0$ $Y_0$ $X_0 + Y_0$ $X_0Y_0$ Zeroth Witt polynomial is projection onto zeroth coordinate; pointwise operations mean this is a homomorphism.
1 $X_0^p + pX_1$ $Y_0^p + pY_1$ $X_0^p + Y_0^p + p(X_1 + Y_1)$, which is the same as $(X_0 + Y_0)^p + p\left[X_1 + Y_1 + \frac{(X_0^p + Y_0^p) - (X_0 + Y_0)^p}{p}\right]$ $X_0^pY_0^p + pX_0^pY_1 + pY_0^pX_1 + p^2X_1Y_1$, which is the same as $(X_0Y_0)^p + p(X_0^pY_1 + Y_0^pX_1 + pX_1Y_1)$ Things work out.

### Truncations to finite length

We can also consider the ring of Witt vectors of length $d$. This is obtained by taking the ring of Witt vectors and projecting it to the first $d$ coordinates $X_0,X_1,X_2,\dots,X_{d-1}$. The operations all have the property that the formula of the $i^{th}$ coordinate of the sum and product involves only vectors $X_j,Y_j, j \le i$, so the ring operations descend to the quotient and we get a ring structure on the quotient.

## Particular cases

Note that $p$ is the prime with respect to which we are defining the Witt polynomials.

Condition on $R$ Conclusion for ring of Witt vectors (infinite) Conclusion for truncation to length $d$
$p$ is invertible in $R$ Isomorphic to the external direct product of countably many copies of $R$ (i.e., sequences with pointwise sum and product), i.e., $R^{\omega}$ Isomorphic to $R^d$
$R$ is the finite prime field $\mathbb{F}_p$ Isomorphic to the ring of p-adic integers, where a Witt vector $(X_0,X_1,X_2,\dots)$ is identified with the $p$-adic integer $\sum_{i=0}^\infty X_ip^i$. Isomorphic to the ring of integers mod $p^d$, i.e., the ring $\mathbb{Z}/p^d\mathbb{Z}$.
$R$ is a finite field of order $p^k$, $k$ a natural number Isomorphic to an unramified degree $k$ extension of the ring of p-adic integers. There is only one such extension up to isomorphism. Isomorphic to the Galois ring, which is an unramified extension of $\mathbb{Z}/p^d\mathbb{Z}$ of degree $k$, and can be constructed using any minimal polynomial for $\mathbb{F}_{p^k}$ over $\mathbb{F}_p$.