Deriving the ultrametric from the p-adic norm?

Question

I had thought that the ultra-metric property was just a rule that someone made up, that if applied shows some bizarre behavior. I however came across these notes: Lecture notes and it seems that the ultra-metric property is actually derived from a p-adic norm. i however don't understand what they are doing (Note Theorem 0.4 which leads to Definition 0.5).

So it seems as if the ultra-metric or non-Archimedean metric is actually derived from the Archimedean metric as applied to p-adic numbers?

Thanks,

Brian

Answer 1

you write:

it seems as if the ultra-metric or non-Archimedean metric is actually derived from the Archimedean metric as applied to p-adic numbers?

it would be more appropriate to say that the ultra-metric on $\mathbb{Q}$ is derived from the same definitions as the more intuitive Archimedean metric. as it turns out such metrics satisfy a strictly stronger form of the subadditivity condition than do Archimedean metrics, but this stronger condition does not have to be assumed, it "comes out in the wash", so to speak.

let us recap the definitions.

a norm $||.|| $ defined on a field is a function whose image lies in the non-negative reals. it is satisfies three conditions (e.g. in the reference you cite).

(a) faithful: $||a||=0 \Rightarrow a=0$

(b) subadditive: $||a+b|| \le ||a|| +||b||$

(c) multiplicative $||ab||=||a||\cdot ||b||$

you can easily see that for $\beta \gt 0$ if $||.||$ is a norm, then if $||.||^{\beta}$ - is subadditive it is also a norm. this gives an equivalence relation on norms.

Ostrowski's (1916) theorem gives a complete description of the equivalence classes of norms on $\mathbb{Q}$. it seems a result of some depth, yet its proof does not require anything more than high-school arithmetic. however few high-school students would find the result particularly interesting. it occupies a rather dry corner of real analysis.

as often the case, when examined carefully this dryness conceals something of considerable interest.

conceptually what we find is that there is a difference of type between norms according to their behaviour on the integers of $\mathbb{Q}$.

our untutored intuition of norm is that it measures an idea of "size" which derives from counting. this intuition of size characterizes the Archimedean notion of a metric. that means we would expect a 'larger' number to have a larger norm than a 'smaller' number.

it turns out that the crucial feature of an Archimedean norm is that if $a$ is a positive integer other than $0$ or $1$, then $$ ||a|| \gt 1 $$

the subtlety is that the norm axioms as stated have models in which, for all rational integers $n$ we have: $$ ||n|| \le 1 $$ this seems counterintuitive at first, and the properties of these non-Archimedean norms also seem counter-intuitive. but this only restates the fact that our initial idea of what a norm is do not exhaust the possibilities expressed in the three axioms.

to think about non-Archimedean norms it is useful to jettison the naive idea that they measure size.

Answer 2

This has nothing to do with the usual Archimedean/Euclidean metric. We can define a norm (hence a metric) on $\mathbb{Q}$ as follows: let $\| a \|_p$ be $c^{-\nu_p (a)}$. Here $\nu_p (a)$ is the largest number of powers of $p$ dividing $a$ (possibly negative, or $\infty$ for $a=0$). So things are small exactly when they are highly divisible by $p$

This makes $\mathbb{Q}$ into an example of an ultrametric space, because the number of powers of $p$ dividing $a+b$ is always at least the the number of powers of $p$ that divide both $a$ and $b$.

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Ultrametric spaces are more common than you might think, though we are not used to noticing them. For example, it is a good idea to think of genetic distance as an ultrametric. Fractals, or data that is tree-like, will tend to have ultrametric qualities.

Again, nothing to do with the Archimedean norm—though it turns out that, up to scaling, the only possible norms on $\mathbb{Q}$ are the Archimedean one, and the $p$-adic ones, a fact of great interest to number theorists.

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Normed fields that do not have the ultrametric property are actually quite rare: they are just the subfields of $\mathbb{C}$. On the other hand, there are ultrametric fields of every infinite cardinality.

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I think the $p$-adics are not such a great first example for an ultrametric space, or even for a normed field. For the former, any discrete space will do. For the latter, we could look instead at the ring $\mathbb{C}(T)$ of rational functions, and measure divisibility by $T$ instead of $p$ (and otherwise mimic the $p$-adics, as $\mathbb{C}[T]$ and $\mathbb{Z}$ are much more similar than they first appear...).

I think this might be a little more intuitive, because it's about measuring the order of poles and zeros of meromorphic functions, which feels more natural, at least at first, than measuring the number of powers of $p$ dividing a number*.

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*My dear departed teacher, Paul Sally, claimed to have accidentally referred to this, during a lecture, as "the $p$-ness of a number".