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 is a function satisfying the following axioms:

- if and only if
- (triangle inequality)

for all .

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

**Definition:** The absolute value is Archimedean if, for , , for some natural number .

Clearly the standard absolute value (which is defined on and , and therefore ) is Archimedean. But wait: since we assumed , we can divide both sides by to obtain . In other words, we can write the definition equivalently as:

**Equivalent Definition:** The absolute value is Archimedean if, for all , for some natural number .

Here takes the place of . The important thing here is that can be *any* element of 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 for all nonzero ). Thus, for some , . So, either or . Thus by taking arbitrarily high powers of or , we can obtain arbitrarily high absolute values. So we can reformulate the definition as follows:

**Equivalent Definition:** is Archimedean if the set contains arbitrarily large elements.

In other words, the set is unbounded. So, is non-Archimedean if the sequence is bounded. However, if any , then taking arbitrarily high powers of can give us arbitrarily high absolute values. So

**Equivalent Definition:** is non-Archimedean if for .

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

**Theorem/Equivalent Definition:** is non-Archimedean if .

*Proof*: (to be added)