properties of non-archimedean valuations

If K is a field, and || a nontrivial non-archimedean valuation (or absolute valueMathworldPlanetmathPlanetmathPlanetmath) on K, then || has some properties that are counterintuitive (and that are false for archimedean valuations).

Theorem 1.

Let K be a field with a non-archimedean absolute value ||. For r>0 a real number, xK, define

B(x,r)={yK|x-y|<r},the open ball of radius r at x
B¯(x,r)={yK|x-y|r},the closed ball of radius r at x


  1. 1.

    B(x,r) is both open and closed;

  2. 2.

    B¯(x,r) is both open and closed;

  3. 3.

    If yB(x,r) (resp. B¯(x,r)) then B(x,r)=B(y,r) (resp. B¯(x,r)=B¯(y,r));

  4. 4.

    B(x,r) and B(y,r) (resp. B¯(x,r) and B¯(y,r)) are either identical or disjoint;

  5. 5.

    If B1=B(x,r) and B2=B(y,s) are not disjoint, then either B1B2 or B2B1;

  6. 6.

    If (xn) is a sequence of elements of K with limnxn=0, then n=1xn is Cauchy (and thus if K is completePlanetmathPlanetmath, a sufficient condition for convergence of a series is that the terms tend to zero)

Proof.  We start by proving (3). Suppose yB(x,r). If zB(x,r), then since the absolute value is non-archimedean, we have


so that zB(y,r). Clearly xB(y,r), so reversing the roles of x and y, we see that B(x,r)=B(y,r). Finally, replacing B by B¯ and < by , we get equality of closed ballsPlanetmathPlanetmath as well.

(4) is now trivial: If B(x,r)B(y,r), choose zB(x,r)B(y,r); then by (3), B(x,r)=B(z,r)=B(y,r). An identical argument proves the result for closed balls.

To prove (5), choose zB1B2. Assume first that rs; then B(z,r)=B1, and B(z,r)B(z,s)=B2, so that B1B2. If sr, then we have identically that B2B1. (Note that (4) is a special case when r=s).

(1) and (2) now follow: for (1), note that B(x,r) is obviously open; its complement consists of a union of open balls of radius r disjoint with B(x,r) and its complement is therefore open. Thus B(x,r) is closed. For (2), B¯(x,r) is obviously closed; to see that it is open, take any yB¯(x,r); then B¯(x,r)=B¯(y,r) and thus B(y,s)B¯(y,r) for s<r is an open neighborhood of y contained in B¯(x,r), which is therefore open.

Finally, to prove (6), we must show that given ϵ, we can find N>0 sufficiently large such that |i=mnxi|<ϵ whenever m,n>N. Simply choose N such that |xi|<ϵ for i>N; then

Title properties of non-archimedean valuations
Canonical name PropertiesOfNonarchimedeanValuations
Date of creation 2013-03-22 18:01:11
Last modified on 2013-03-22 18:01:11
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic CompleteUltrametricField