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

Yoshifumi Tsuchimoto

No.02: % latex2html id marker 608
\fbox{definition of $\mathbb {Z}_p$}

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

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

    \begin{displaymath}
% latex2html id marker 707\vert n\vert _p=
\begin{cases}
...
...{ and } p^{k+1}\not \vert n \\
0 & \text{ if } n=0
\end{cases}\end{displaymath}

  2. We define a $ p$ -adic distance $ d_p(\bullet,\bullet)$ on $ \mathbb{Z}$ as follows.

    % latex2html id marker 715
$\displaystyle 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. % latex2html id marker 724
$ \vert x\vert _p\geq 0\quad (\forall x \in \mathbb{Z}).\qquad \vert x\vert _p=0 \iff x=0.$
    2. % latex2html id marker 726
$ \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

    $\displaystyle +: \mathbb{Z}_p\times \mathbb{Z}_p \to \mathbb{Z}_p
$

    (addition) and

    $\displaystyle \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

$\displaystyle 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

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

EXERCISE 02.1   compute

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


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Yoshifumi Tsuchimoto 2016-04-15