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I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions), and came across the exercise:

Prove that a norm $\|.\|$ on a field $F$ is non-Archimedean if and only if $$\{x\in F : \|x\| < 1 \} \cap \{x\in F : \|x-1\| < 1 \} = \emptyset.$$

In one direction, the proof was trivial, and in the other direction somewhat harder, but my question is really about where this question comes from. If I were trying to think up exercises on this topic, I don't think I would have thought of this one in a million years.

I am (gradually) getting used to the "eccentricities" of non-Archimedean metrics, but if someone could give me some idea of the intuition that lies behind this particular property, I would be grateful.

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$U^1 = \{x \in F : |x-1| < 1\}$ is an important group in number theory. I'm not sure what it's called in English, in German it's Einseinheiten. The norm residue symbol $(-,E/F)$ maps it surjectively onto the ramification group of $E/F$. It's a subgroup of the units of $\mathcal{O}_F$ such that $\mathcal{O}_F^{\times} / U^1 \cong \kappa^{\times}$ where $\kappa$ is the remainder field. Also, by local class field theory it's contained in the norm group $N_{E/F}E^{\times}$ if and only if $E/F$ is tamely ramified. – Cocopuffs Jul 17 '12 at 11:28
Wow. It might take me a while to digest that! Thanks. – Old John Jul 17 '12 at 11:31
So the intuition I think of this problem is to see that such one-units are really units in the normal since. – Cocopuffs Jul 17 '12 at 11:35
Perhaps the down-voter would be good enough to give me a clue as to what is wrong with my question? – Old John Dec 5 '13 at 18:48
up vote 5 down vote accepted

The defining property of ultrametrics is that in every triangle two longer sides are equal: more precisely, if ABC is a triangle (=triple of points) and $|AB|\ge |BC|\ge |AC|$ then $|AB|=|BC|$. Now, the exercise asks about the existence of a triangle in which one side has length 1 while the other two are strictly shorter: designed to be a contradiction to the definition.

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Thanks, that makes it very clear (and I am annoyed that I didn't see it in that light myself!). So, would I be right in saying that the points $0$ and $1$ don't really have any special significance here, apart from being two elements that are bound to exist in a field? – Old John Jul 17 '12 at 11:39
@OldJohn It also matters that the distance between them is 1. – user31373 Jul 17 '12 at 11:41

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