$\mathbb{Z}_p$ , $\mathbb{Q}_p$ , and the ring of Witt vectors

Yoshifumi Tsuchimoto

No.02:

Let $p$ be a prime (``base''). We would like to introduce a metric on $\mathbb{Z}$ such that

$$n$$ :small $$\iff n$$    is divisible by powers of $$p.$$

Namely:

DEFINITION 02.1   Let $p$ be a prime number.

1. We define a $p$ -adic norm $\vert\bullet\vert _p$ on $\mathbb{Z}$ as follows.
   $$% latex2html id marker 707\vert n\vert _p= \begin{cases} ... ...{ and } p^{k+1}\not \vert n \\ 0 & \text{ if } n=0 \end{cases}$$

2. We define a $p$ -adic distance $d_p(\bullet,\bullet)$ on $\mathbb{Z}$ as follows.
   $$d_p(n,m)=\vert n-m\vert _p \qquad(n,m\in \mathbb{Z})$$

LEMMA 02.2  

1. $\vert\bullet\vert _d$ enjoys the following properties.
   1. $\vert x\vert _p\geq 0\quad (\forall x \in \mathbb{Z}).\qquad \vert x\vert _p=0 \iff x=0.$
   2. $\vert x+y\vert _p\leq \max (\vert x\vert _p ,\vert y\vert _p) \quad (\leq (\vert x\vert _p + \vert y\vert _p)$ .
   3. $\vert x y\vert _p = \vert x\vert _p\vert y\vert _p$
2. $(\mathbb{Z}, d_p)$ is a metric space.

DEFINITION 02.3   A metric space $(X,d)$ is said to be complete if every Cauchy sequence of $X$ converges to an element of $X$ .

THEOREM 02.4   Let $(X,d)$ be a metric space. There exists a complete metric space $(\bar X,d)$ with an isometry $\iota:X \to \bar X$ such that $X$ is dense in $\bar X$ . Furthermore, $\bar X$ is unique up to a unique isometry.

DEFINITION 02.5   Let $(X,d)$ be a metric space. We call $(\bar X,d)$ as in the above theorem the completion of $(X,d)$ .

DEFINITION 02.6   Let $p$ be a prime number. We denote the completion of $(\mathbb{Z}, d_p)$ by $(\mathbb{Z}_p,d_p)$ and call it the ring of $p$ -addic integers. Thus elements of $\mathbb{Z}_p$ are $p$ -addic integers.

THEOREM 02.7   $\mathbb{Z}_p$ has a unique structure of a topological ring. Namely,

1. There exists unique continuous maps
   $$+: \mathbb{Z}_p\times \mathbb{Z}_p \to \mathbb{Z}_p$$
   (addition) and
   $$\times: \mathbb{Z}_p\times \mathbb{Z}_p \to \mathbb{Z}_p$$
   (multiplication) which are extensions of the usual addition and multiplication of $\mathbb{Z}$ .
2. $(\mathbb{Z}_p,+,\times)$ is a commutative associative ring.

DEFINITION 02.8   Let $p$ be a prime number. For any sequence $\{a_j \}_{j=0}^\infty$ such that $a_j \in \{0,1,2,3,\dots, p-1\}$ , we consider a sequence $\{s_n\}$ defined by

$$s_n=\sum_{j=0}^n a_j p^j.$$

Then the sequence $\{s_n\}$ is a Cauchy sequence in $\mathbb{Z}_p$ . We denote the limit of the sequence as

$$[0.a_0 a_1 a_2 a_3 \dots ]_p.$$

EXERCISE 02.1   compute

$$[0.1]_3+[0.2222\dots]_3$$