# Witt vectors

###### Theorem 1.

Let $p$ be a prime and let $\mathbb{K}$ be a perfect ring of characteristic $p$. There exists a unique strict $p$-ring (http://planetmath.org/PRing) $W(\mathbb{K})$ with residue ring $\mathbb{K}$.

###### Definition 1.

Let $\mathbb{K}$ be a perfect ring of characteristic $p$. The unique strict $p$-ring (http://planetmath.org/PRing) $W(\mathbb{K})$ with residue ring $\mathbb{K}$ is called the ring of Witt vectors with coefficients in $\mathbb{K}$.

Next, we give an explicit construction of the Witt vectors.

###### Definition 2.

Let $p$ be a prime number  and let $\{X_{i}\}_{i=0}^{\infty}$ be a sequence of indeterminates. The polynomials $W_{n}\in\mathbb{Z}[X_{1},\ldots,X_{n}]$ given by:

 $\displaystyle W_{0}$ $\displaystyle=$ $\displaystyle X_{0},$ $\displaystyle W_{1}$ $\displaystyle=$ $\displaystyle X_{0}^{p}+pX_{1},$ $\displaystyle W_{n}$ $\displaystyle=$ $\displaystyle X_{0}^{p^{n}}+pX_{1}^{p^{n-1}}+\ldots+p^{n}X_{n}=\sum_{i=0}^{n}p% ^{i}X_{i}^{p^{n-i}}.$

are called the Witt polynomials.

###### Proposition 1.

Let $\{X_{i}\},\ \{Y_{i}\}$ be two sequences of indeterminates. For every polynomial in two variables $Q(U,V)\in\mathbb{Z}[U,V]$ there exist polynomials $\{t_{i}\}_{i=0}^{\infty}$ in the variables $\{X_{i}\}$ and $\{Y_{i}\}$

 $t_{i}\in\mathbb{Z}[\{X_{i}\},\{Y_{i}\}]$

such that

 $W_{n}(t_{0},t_{1},t_{2},\ldots,t_{n})=Q(W_{n}(X_{0},X_{1},\ldots),W_{n}(Y_{0},% Y_{1},\ldots))$

for all $n\geq 0$.

###### Proof.

See , p. 40. ∎

Let $S_{0},\ S_{1},\ S_{2},\ldots$ (resp. $P_{0},\ P_{1},\ P_{2},\ldots$) be the polynomials $t_{0},\ t_{1},\ t_{2},\ldots$ associated with $Q(U,V)=U+V$ (resp. $Q(U,V)=U\cdot V$) given by the previous proposition. We will use the polynomials $S_{i}$, $P_{i}$ to define the addition and multiplication in a new ring. In the following proposition, the notation $R^{\infty}$ stands for the set of all sequences $(r_{1},r_{2},\ldots)$ of elements in $R$.

###### Theorem 2.

Let $\mathbb{K}$ be a perfect ring of characteristic $p$. We define a ring $W=(\mathbb{K}^{\infty},+,\cdot)$ where the addition and multiplication, for $k,h\in\mathbb{K}^{\infty}$, are defined by:

 $k+h=(S_{0}(k,h),S_{1}(k,h),\ldots),\quad k\cdot h=(P_{0}(k,h),P_{1}(k,h),% \ldots).$

Then the ring $W$ concides with $W(\mathbb{K})$, the ring of Witt vectors with coefficients in $\mathbb{K}$.

###### Definition 3.

Let $\mathbb{K}$ be a perfect ring of characteristic $p$. We define the ring of Witt vectors of length $n$ (over $\mathbb{K}$) to be the ring $W_{n}(\mathbb{K})=(\mathbb{K}^{n-1},+,\cdot)$, where, for $k,h\in\mathbb{K}^{n-1}$:

 $k+h=(S_{0}(k,h),\ldots,S_{n-1}(k,h)),\quad k\cdot h=(P_{0}(k,h),\ldots,P_{n-1}% (k,h)).$

It is clear from the definitions that:

 $W(\mathbb{K})=\varprojlim W_{n}(\mathbb{K})$

In words, $W(\mathbb{K})$ is the projective limit of the Witt vectors of finite length.

###### Example 1.

Let $\mathbb{K}=\mathbb{F}_{p}$. Then $W_{n}(\mathbb{F}_{p})=\mathbb{Z}/p^{n}\mathbb{Z}$. Thus:

 $W(\mathbb{F}_{p})=\mathbb{Z}_{p},$

the ring of $p$-adic integers (http://planetmath.org/PAdicIntegers).

## References

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Title Witt vectors WittVectors 2013-03-22 15:14:31 2013-03-22 15:14:31 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 13K05 msc 13J10 Witt polynomials