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

No.06: \fbox{$\mathbb {Q}_p$ }

DEFINITION 06.1   We denote by $ \mathbb{Q}_p$ the quotient field of $ \mathbb{Z}_p$ .

LEMMA 06.2   Every non zero element $ x \in\mathbb{Q}_p$ is uniquely expressed as

% latex2html id marker 619
$\displaystyle x= p^k u \qquad( k\in \mathbb{Z}, u \in \mathbb{Q}_p^\times).
$

We have so far constructed a ring $ \mathbb{Z}_p$ and a field $ \mathbb{Q}_p$ for each prime $ p$ .

PROPOSITION 06.3   Let $ p$ be a prime. Then:
  1. $ \mathbb{Z}_p$ is a local ring with the unique maximal ideal $ p \mathbb{Z}_p$ .
  2. $\displaystyle \mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p (=\mathbb{Z}/p \mathbb{Z}).
$

  3. $ \mathbb{Z}_p$ is an integral domain whose quotient field $ \mathbb{Q}_p$ is a field of characteristic zero.

With $ \mathbb{Q}_p$ and/or $ \mathbb{Z}_p$ , we may do some ``calculus'' such as:

THEOREM 06.4   [1, corollary 1 of theorem 1] Let $ f\in \mathbb{Z}_p[X_1,X_2,\dots,X_m], x\in \mathbb{Z}_p^m$ , $ n,k\in \mathbb{Z}$ . Assume that there exists a natural number $ j$ such that % latex2html id marker 659
$ 1\leq j \leq m$ ,

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$\displaystyle \frac{\partial f}{\partial X_j} (x)\not \equiv 0 \pmod p.
$

Then there exists $ y\in \mathbb{Z}_p^m $ such that

(1)   $\displaystyle f(y)=0$
(2)   % latex2html id marker 665
$\displaystyle y\equiv x \pmod p$

See [1] for details.


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2008-06-10