A generalization of the p-adic Number first proposed by Kürschák in 1913. A valuation on a Field is a Function from to the Real Numbers such that the following properties hold for all :
If (4) is satisfied for , then satisfies the Triangle Inequality,
The simplest valuation is the Absolute Value for Real Numbers. A valuation satisfying (4b) is called non-Archimedean Valuation; otherwise, it is called Archimedean.
If is a valuation on and
, then we can define a new valuation by
(1) |
If two valuations are equivalent, then they are both non-Archimedean or both Archimedean. , , and with the usual Euclidean norms are Archimedean valuated fields. For any Prime , the p-adic Number with the -adic valuation is a non-Archimedean valuated field.
If is any Field, we can define the trivial valuation on by for all and , which is a non-Archimedean valuation. If is a Finite Field, then the only possible valuation over is the trivial one. It can be shown that any valuation on is equivalent to one of the following: the trivial valuation, Euclidean absolute norm , or -adic valuation .
The equivalence of any nontrivial valuation of
to either the usual Absolute Value or to a p-adic Number absolute value was proved by Ostrowski
(1935). Equivalent valuations give rise to the same topology. Conversely, if two valuations have the same topology, then
they are equivalent. A stronger result is the following: Let , , ..., be valuations
over which are pairwise inequivalent and let , , ..., be elements of . Then there exists an
infinite sequence (, , ...) of elements of such that
(2) |
(3) |
(4) |
A discrete valuation is a valuation for which the Valuation Group is a discrete subset of the Real
Numbers . Equivalently, a valuation (on a Field ) is discrete if there exists a Real
Number such that
(5) |
If is a valuation on , then it induces a metric
(6) |
See also Absolute Value, Local Field, Metric Space, p-adic Number, Strassman's Theorem, Ultrametric, Valuation Group
References
Cassels, J. W. S. Local Fields. Cambridge, England: Cambridge University Press, 1986.
Ostrowski, A. ``Untersuchungen zur aritmetischen Theorie der Körper.'' Math. Zeit. 39, 269-404, 1935.
© 1996-9 Eric W. Weisstein