In this post, we will see that the image of $\mathbb{C}_p$, the complete algebraic closure of $\mathbb{Q}_p$, has a non-complete image in $\mathbb{R}$. To do so, let’s briefly review what the $p$-adic numbers are.

1. $p$-adic Numbers

Throughout this post, $p$ is a fixed positive prime number. The $p$-adic valuation of an integer $n$ is defined to be

\[\nu_p(n) \coloneqq \begin{cases} \max\left\{ k \in \mathbb{Z}_{\ge 0} : p^k \mid n \right\} &\text{if } n \neq 0, \\ \infty &\text{if } n = 0, \end{cases}\]

where $\mathbb{Z}_{\ge 0}$ denotes the set of non-negative integers. For a rational number $r/s$ with $r \in \mathbb{Z}$ and $s \in \mathbb{Z} \setminus \{0\}$, we define

\[\nu_p\left( \frac{r}{s} \right) \coloneqq \nu_p(r) - \nu_p(s).\]

Then, the function $\nu_p : \mathbb{Q} \to \mathbb{Z} \cup \{\infty\}$ satisfies

  1. $\nu_p(xy) = \nu_p(x) + \nu_p(y)$
  2. $\nu_p(x+y) \ge \min\{ \nu_p(x), \nu_p(y) \}$ (equality holds when $\nu_p(x) \neq \nu_p(y)$)
  3. $\nu(x) = \infty \iff x = 0$

(i.e., $\nu_p$ is a discrete valuation on $\mathbb{Q}$). The $p$-adic absolute value on $\mathbb{Q}$ is the function $|\cdot|_p : \mathbb{Q} \to \mathbb{R}_{\ge 0}$ defined by

\[|x|_p \coloneqq p^{-\nu_p(x)}.\]

It is an easy exercise to show that $|\cdot|_p$ satisfies non-negativity, positive-definiteness, multiplicativity, and the triangle inequality. In fact, it satisfies an inequality stronger than the triangle inequality, called strong traingle inequality or ultrametric inequality

\[|x+y|_p \le \max \left\{ |x|_p, |y|_p\right\},\]

where the equality holds when $|x|_p \neq |y|_p$ Thus, this absolute value induces a metric $d : \mathbb{Q} \times \mathbb{Q} \to \mathbb{R}_{\ge 0}$ defined by $d(x,y)\coloneqq |x-y|_p$. The completetion with respet to this metric to the $\mathbb{Q}$ is $\mathbb{Q}_p$, the $p$-adic numbers.


2. Theory of Non-Archimedean Fields.

In this section, we introduce non-Arhimdean fields, which are generalizations of $p$-adic numbers, and review their important properties.

Definition. Let $K$ be a field. A map $|\cdot|: K \to \mathbb{R}_{\ge 0}$ is called a non-Archimedean absolute value if for all $x,y \in K$ the following hold:

  1. $|x| = 0 \iff x = 0$,
  2. $|xy| = |x||y|$,
  3. $|x+y| \le \max\{|x|, |y|\}$

A fiedl with a non-Archimdean absolute value is called a non-Archimedean field.

Lemma 1. Let $K$ be a field with a non-Archimedean absolute value $|\cdot|$. For $x,y \in K$ with $|x| \neq |y|$, we have $|x+y| = \max\{ |x|, |y| \}$.

Proof. Without loss of generality, assume $|x| < |y|$. Then we get $|x+y| \le \max\{|x|, |y|\} = |y|$. On the other hand, we have

\[|y| = |(x+y)-x| \le \max\{ |x+y|, |x| \},\]

so $|y| \le |x+y|$ by the assumption that $|x| < |y|$. This completes the proof. $\quad\square$

Let $K$ be a field with a non-Archimedean absolute value $|\cdot|$. As usual, the absolute value gives a metric $d(x,y) \coloneqq |x-y|$. A non-Archimedan field $K$ is called complete if every Cauchy sequence converges in $k$. For instance, the $p$-adic number $\mathbb{Q}_p$ is a typical example of complete non-Archimedean fields. Complete non-Archimedean field has several useful algebraic properties.

Theroem 1. Let $K$ be a complete with respect to the given non-archimedean absolute value $|\cdot|$, and let $L$ be an algebraic extension of $K$. Then there exists a unique non-Archimedean absolute value on $L$ extending the absolute value $|\cdot|$ from $K$. Furthermore, $L$ is complete if $[L : K] < \infty$.

Proof. See Theorem 2 of (3.2.4) of [1].$\quad\square$

Proposition 1. Let $K$ be complete with respect to the given non-Archimedean absolute value $|\cdot|$, and let $L$ be a finite extension of $K$. For $y \in L$, let $X^m + a_{m-1}X^{m-1} + \cdots a_0 \in K[X]$ be the minimal monic polynomial of $y$ over $K$. Then

\[|y| = |a_m|^{1/m} = \left| N_{L/K}(y) \right|^{1/[L:K]},\]

where $N_{L/K}: L \to K$ is the field norm.

Proof. See Proposition 3 of (3.2.4) of [1].$\quad\square$

For a field $K$ provided with a non-archimedean absolute value $|\cdot|$, we denote its algebraic closure by $\overline{K}$, and we denote its completion by $\hat{K}$.

Proposition 2. Let $K$ be a non-Archimedean field. If $K$ is algebraically closed, then completion $\widehat{K}$ of $K$ is algebraically closed.

Proof. See Proposition 3 of (3.4.1) of [1].$\quad\square$


3. Algebraically Closed Complete Field $\mathbb{C}_p$

We denote the completion of the algebraically closure of $\mathbb{Q}_p$ by $\mathbb{C}_p$. Then by the Proposition 2, the field $\mathbb{C}_p$ has a unique extension of $|\cdot|_p$, and it is still algebraically closed. For a non-Archimedean field $K$ with an absolute value $|\cdot| : K \to \mathbb{R}_{\ge 0}$, we denote the image of the function $|\cdot|$ by $|K|$, which is a subset of $\mathbb{R}$. In this section, we calculate what $|\mathbb{C}_p|$ exactly is. To do so, we need an (almost trivial) claim, which is a result of ultrametric inequality.

Claim. Let $K$ be a field with a non-Archimedean absolute value $|\cdot|$. Then $|K| = \left|\widehat{K} \right|$.

Proof. Let $x$ be a nonzero element of $\widehat{K}$. Then there exists a Cauchy sequence $\{x_i\}_i$ in $K$ such that $x_i \to x$. We may assume that $x_i \neq 0$ for all $i$. Since $\{x_i\}_i$ is Cauchy, and since $\{|x_i|\}$ must be bounded below, there exists some sufficiently large positive integer $N$ such that for all $m \ge n \ge N$

\[|x_n - x_m| < |x_n|.\]

Then the ultrametric triangular inequality tells us that

\[|x_m| = |x_n + (x_m - x_n)| = \max\{ |x_n|, |x_m - x_n| \} = |x_n|.\]

This shows that $|x| = |x_N| \in |K|$. $\quad\square$

By the definition, we have $|\mathbb{Q}_p| = p^{\mathbb{Z}} \cup \{0\}$ where $p^{\mathbb{Z}} \coloneqq \{ p^n : n \in \mathbb{Z} \}$. Combining this fact with Propositoin 1, we obtain $|\overline{\mathbb{Q}_p}| = p^{\mathbb{Q}}$ where $p^{\mathbb{Q}} \coloneqq \{ p^r : r \in \mathbb{Q} \}$. However, the Claim shows that $|\mathbb{C}_p| = p^{\mathbb{Q}} \cup {0}$, and it is easy to observe the it is a non-complete subset of $\mathbb{R}$.

I believe that these peculiar topological behaviors of $\mathbb{Q}_p$ and $\mathbb{C}_p$ are related to their total disconnectedness. However, their less intuitive topological behaviors make room for rich algebraic techniques used for studying them.


References

  1. Bosch, S., Güntzer, U., & Remmert, R. Non‑Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry. Grundlehren der mathematischen Wissenschaften 261. Springer‑Verlag, 1984. ISBN 978‑3‑540‑12546‑4 https://link.springer.com/book/9783540125464