properties of non-archimedean valuations
If K is a field, and |⋅| a nontrivial non-archimedean valuation (or absolute value) 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, x∈K, define
B(x,r)={y∈K∣|x-y|<r},the open ball of radius r at x | ||
ˉB(x,r)={y∈K∣|x-y|≤r},the closed ball of radius r at x |
Then
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1.
B(x,r) is both open and closed;
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2.
ˉB(x,r) is both open and closed;
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3.
If y∈B(x,r) (resp. ˉB(x,r)) then B(x,r)=B(y,r) (resp. ˉB(x,r)=ˉB(y,r));
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4.
B(x,r) and B(y,r) (resp. ˉB(x,r) and ˉB(y,r)) are either identical or disjoint;
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5.
If B1=B(x,r) and B2=B(y,s) are not disjoint, then either B1⊂B2 or B2⊂B1;
- 6.
Proof. We start by proving (3). Suppose . If , then since the absolute value is non-archimedean, we have
so that . Clearly , so reversing the roles of and , we see that . Finally, replacing by and by , we get equality of closed balls as well.
(4) is now trivial: If , choose ; then by (3), . An identical argument proves the result for closed balls.
To prove (5), choose . Assume first that ; then , and , so that . If , then we have identically that . (Note that (4) is a special case when ).
(1) and (2) now follow: for (1), note that is obviously open; its complement consists of a union of open balls of radius disjoint with and its complement is therefore open. Thus is closed. For (2), is obviously closed; to see that it is open, take any ; then and thus for is an open neighborhood of contained in , which is therefore open.
Finally, to prove (6), we must show that given , we can find sufficiently large such that whenever . Simply choose such that for ; then
Title | properties of non-archimedean valuations |
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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 |