Archimedean absolute values | Arithmetic variety

In the previous post we discussed the Archimedean property for an ordered field. Today I’ll discuss the Archimedean property for valued fields, that is, fields equipped with an absolute value.

Recall that an absolute value on a field $K$ is a function $| \ |: K \to \mathbb{R}_{\geq 0}$ satisfying the following axioms:

$|a| = 0$ if and only if $a = 0$
$|ab| = |a||b|$
$|a+b| \leq |a| + |b|$ (triangle inequality)

for all $a, b \in K$ .

Here is an intuitive, analogous definition for the Archimedean property:

Definition: The absolute value $| \ |$ is Archimedean if, for $x, y \in K$ , $x \neq 0$ , $|nx| > |y|$ for some natural number $n$ .

Clearly the standard absolute value (which is defined on $\mathbb{C}$ and $\mathbb{R}$ , and therefore $\mathbb{Q}$ ) is Archimedean. But wait: since we assumed $x \neq 0$ , we can divide both sides by $|x|$ to obtain $|n| > |y/x|$ . In other words, we can write the definition equivalently as:

Equivalent Definition: The absolute value $| \ |$ is Archimedean if, for all $a \in K$ , $|n| > |a|$ for some natural number $n$ .

Here $a$ takes the place of $y/x$ . The important thing here is that $a$ can be any element of $K.$ So what this is saying is that, given any element of the field, there is some natural number that beats it.

Now, let us assume that the absolute value is nontrivial. (The trivial absolute value has $|a| = 1$ for all nonzero $a$ ). Thus, for some $a$ , $|a| \neq 1$ . So, either $|a| > 1$ or $|1/a| = 1/|a| > 1$ . Thus by taking arbitrarily high powers of $a$ or $1/a$ , we can obtain arbitrarily high absolute values. So we can reformulate the definition as follows:

Equivalent Definition: $| \ |$ is Archimedean if the set $\{ |n| \ | \ n \in \mathbb{N}\}$ contains arbitrarily large elements.

In other words, the set is unbounded. So, $| \ |$ is non-Archimedean if the sequence $|1|, |2|, |3|, \dots$ is bounded. However, if any $|n| > 1$ , then taking arbitrarily high powers of $n$ can give us arbitrarily high absolute values. So

Equivalent Definition: $| \ |$ is non-Archimedean if $|n| \leq 1$ for $n \in \mathbb{N}$ .

Finally, I will present another very useful characterization of the (non)Archimedean property.

Theorem/Equivalent Definition: $| \ |$ is non-Archimedean if $|a+b| \leq \text{max}(|a|, |b|)$ .

Proof: (to be added)

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