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In this entry we define a commutative ring, the Witt vectors, which is particularly useful in number theory, algebraic geometry and other areas of commutative algebra. The Witt vectors are named after Ernst Witt.
Definition 1 Let $\mathbb{K}$ be a perfect ring of characteristic $p$ . The unique strict $p$ -ring $W(\mathbb{K})$ 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 $W_n\in\Ints[X_1,\ldots,X_n]$ given by: \begin{eqnarray*} W_0 &=& X_0,\\ W_1 &=& X_0^p+pX_1,\\ W_n &=& X_0^{p^n}+pX_1^{p^{n-1}}+\ldots+p^nX_n=\sum_{i=0}^n p^iX_i^{p^{n-i}}. \end{eqnarray*}are called the Witt polynomials.
Proposition 1 Let $\{X_i\},\ \{Y_i\}$ be two sequences of indeterminates. For every polynomial in two variables $Q(U,V)\in \Ints[U,V]$ there exist polynomials $\{t_i\}_{i=0}^\infty$ in the variables $\{X_i\}$ and $\{Y_i\}$ $$t_i \in \Ints[\{X_i\},\{Y_i\}]$$ 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 [ 1], 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 $W_n(\mathbb{F}_p)=\Ints/p^n\Ints$ . Thus: $$W(\mathbb{F}_p)=\Ints_p,$$ the ring of $p$ -adic integers.
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- J. P. Serre, Local Fields, Springer-Verlag, New York.
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