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

No.06:

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

$$x= p^k u \qquad( k\in \mathbb{Z}, u \in \mathbb{Q}_p^\times).$$

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. $$\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p (=\mathbb{Z}/p \mathbb{Z}).$$

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 $1\leq j \leq m$ ,

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

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

$$f(y)=0$$ (1)

$$y\equiv x \pmod p$$ (2)

See [1] for details.