Absolute Values

We start by defining absolute values on $\mathbb Q$, but much of this extends naturally to absolute values on other fields. An absolute value on $\mathbb Q$ is a function $\mathbb Q \rightarrow [0,\infty)$ such that:

1. positive definiteness: $|x| = 0$ iff $x = 0$,
2. multiplicativity: $|x y| = |x| |y|$ for all $x, y \in \mathbb Q$
3. triangle inequality: $|x + y| \leq |x| + |y|$ for all $x,y \in \mathbb Q$.

If, in place of (3), $|\cdot|$ satisfies the strong triangle inequality: $|x + y| \leq \max\{|x|, |y|\}$, then we say $| \cdot |$ is a non-archimedean absolute value.

Example

Let $p$ be a prime, and define the standard $p$-adic absolute on $\mathbb Q$ by
$$\left|p^{\ell} \frac{a}b\right| = p^{-\ell}.$$
Here we assume that $a$ and $b$ are rational integers not divisible by $p$. $| \cdot|_p$ is a non-archimedean absolute value. Note that if $n$ is a rational integer, then $|n| \leq 1$.

Completions

Given an absolute value $|\cdot|$ on $\mathbb Q$, we say a sequence $(x_n)$ is Cauchy if given $\epsilon > 0$, there exists $N \in \mathbb N$ so that $n, m > N$ implies $|x_n – x_m| < \epsilon$. Similarly, we say that $(x_n)$ converges to zero if given $\epsilon > 0$ there exists $N \in \mathbb N$ so that $n > N$ implies $|x_n| < \epsilon$. Using the standard operations on sequences, the set of Cauchy sequences $\mathcal C$ is a (commutative) ring, and the set of sequences which converge to zero, $\mathcal Z$ is a maximal ideal. It follows that $\mathcal C / \mathcal Z$ is a field, and we call this the completion of $\mathbb Q$ with respect to the absolute value $| \cdot |$. We may embed $\mathbb Q$ into $\mathcal C / \mathcal Z$ by the diagonal embedding $x \mapsto (x,x,\cdots)$. The fact that $\mathbb Q$ is dense in $\mathcal C / \mathcal Z$ allows us to extend $| \cdot |$ to a unique absolute value on $\mathcal C / \mathcal Z$.

The completion of $\mathbb Q$ with respect to the usual absolute value is $\mathbb R$.
The completion of $\mathbb Q$ with respect to $| \cdot |_p$ is called the field of $p$-adic numbers and denoted $\mathbb Q_p$.

The Topology of $\mathbb Q_p$

A basic neighborhood in $\mathbb Q_p$ is a set of the form $B(x, \epsilon) = \{y \in \mathbb Q_p : |y – x|_p \leq \epsilon\}$. Since $| \cdot |_p$ takes values in the discrete set $\{p^{-n} : n \in \mathbb Z\}$, every basic neighborhood is both open and closed, since for instance
$$\{y : |y|_p \leq p^{-n} \} = \{y : |y|_p < 2p^{-n} \}.$$

Let’s investigate the ball $B(1, p^{-1})$ by looking at what integers it contains. Clearly $1 \in B(1, p^{-1})$. The condition $|y – 1| \leq p^{-1}$ is equivalent to $y \equiv 1 \bmod p$. Similarly, $B(1, p^{-n})$ contains all the integers $y$ such that $y \equiv 1 \bmod p^n$. For instance, when $p = 7$, the integers $1$ and $344$ are close, a distance of $7^{-3}$ from each other, since $344 – 1 = 343 = 7^3$.

For a slightly more complicated ball, consider $B(1/2, p^{-2})$. This is by definition
$$\{y \in \mathbb Q_p : |y – 1/2|_p \leq p^{-2} \}.$$
From the definition of $| \cdot |_p$, if $y$ is a rational number so that $|y – 1/2|_p = p^2$, then there exist integers $a$ and $b$, both relatively prime to $p$, so that $y – 1/2 = p^2 a/b$. More generally, if $|y – 1/2|_p \leq p^2$ then there exist integers $a$ and $b$, with $b$ relatively prime to $p$, so that $y – 1/2 = p^2 a/b$. It follows that the rational numbers in $B(1/2, p^{-2})$ are of the form $1/2 + p^2 r$ where $r$ is a rational number which contains no factors of $p$ in the denominator when written in lowest terms.

The General Setup

Suppose $K$ is a number field of degree $d$ over $\mathbb Q$ and suppose $\mf O$ is the ring of integers of $K$. Given a prime ideal $\mf p \subseteq \mf O$ we can define an absolute value $| \cdot |_{\mf p}$ by fixing $q > 0$ and setting
$|x|_{\mf p} = q^{-n} \qquad \mbox{where} \qquad x \in \mf p^n \setminus \mf p^{n+1}.$
The completion of $K$ with respect to $| \cdot |_{\mf p}$ is denoted $K_{\mf p}$.

The ring of integers in $K_{\mf p}$ is defined to be $\mf o_{\mf p} = \{x \in K_{\mf p} : |x|_{\mf p} \leq 1\}$. The ideal $\mf m_{\mf p} \subseteq \mf o_{\mf p}$ given by $\mf m_{\mf p} = \{x \in \mf o_{\mf p} : |x| < 1 \}$. The units in $\mf o_{\mf p}$ are given by $U_{\mf p} := \{x \in \mf o_{\mf p} : |x|_{\mf p} = 1 \}$.

The field $\mf o_{\mf p} / \mf m_{\mf p}$ is a finite field of size $p^f$ for some rational prime $p$ and positive integer $f.$ It is convenient to take $q$ in the definition of $| \cdot |_{\mf p}$ to be $q := p^f.$

Often times, when we are working with a fixed non-archimedean absolute value, we drop the subscript $\mf p$ from various quantities. When in this situation we assume $(K, |\cdot|)$ is a field complete with respect to $|\cdot|$ and with integers $\mf o$, maximal ideal $\mf m$ and units $U$. Unless specifically indicated otherwise we take $q = [\mf o : \mf m]$. The maximal ideal is principal, and we set $\varpi$ to be a generator or uniformizer for the ideal: $\mf m = \varpi \mf o$. Clearly then $|\varpi| = 1/q$.

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