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Valuation


A generalization of the p-adic norm first proposed by Kürschák in 1913. A valuation |·| on a field K is a function from K to the real numbers R such that the following properties hold for all x,y in K:

1. |x|>=0,

2. |x|=0 iff x=0,

3. |xy|=|x||y|,

4. |x|<=1 implies |1+x|<=C for some constant C>=1 (independent of x).

If (4) is satisfied for C=2, then |·| satisfies the triangle inequality,

4a. |x+y|<=|x|+|y| for all x,y in K.

If (4) is satisfied for C=1 then |·| satisfies the stronger ultrametric inequality

4b. |x+y|<=max(|x|,|y|).

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 |·|_1 is a valuation on K and lambda>=1, then we can define a new valuation |·|_2 by

 |x|_2=|x|_1^lambda.
(1)

This does indeed give a valuation, but possibly with a different constant C in axiom 4. If two valuations are related in this way, they are said to be equivalent, and this gives an equivalence relation on the collection of all valuations on K. Any valuation is equivalent to one which satisfies the triangle inequality (4a). In view of this, we need only to study valuations satisfying (4a), and we often view axioms (4) and (4a) as interchangeable (although this is not strictly true).

If two valuations are equivalent, then they are both non-Archimedean or both Archimedean. Q, R, and C with the usual Euclidean norms are Archimedean valuated fields. For any prime p, the p-adic numbers Q_p with the p-adic valuation |·|_p is a non-Archimedean field.

If K is any field, we can define the trivial valuation on K by |x|=1 for all x!=0 and |0|=0, which is a non-Archimedean valuation. If K is a finite field, then the only possible valuation over K is the trivial one. It can be shown that any valuation on Q is equivalent to one of the following: the trivial valuation, Euclidean absolute norm |·|, or p-adic valuation |·|_p.

The equivalence of any nontrivial valuation of Q to either the usual absolute value or to a p-adic norm 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 |·|_1, |·|_2, ..., |·|_k be valuations over K which are pairwise inequivalent and let a_1, a_2, ..., a_k be elements of K. Then there exists an infinite sequence (x_1, x_2, ...) of elements of K such that

 lim_(n->infty w.r.t. |·|_1)x_n=a_1
(2)
 lim_(n->infty w.r.t. |·|_2)x_n=a_2,
(3)

etc. This says that inequivalent valuations are, in some sense, completely independent of each other. For example, consider the rationals Q with the 3-adic and 5-adic valuations |·|_3 and |·|_5, and consider the sequence of numbers given by

 x_n=(43·5^n+92·3^n)/(3^n+5^n).
(4)

Then x_n->43 as n->infty with respect to |·|_3, but x_n->92 as n->infty with respect to |·|_5, illustrating that a sequence of numbers can tend to two different limits under two different valuations.

A discrete valuation is a valuation for which the valuation group is a discrete subset of the real numbers R. Equivalently, a valuation (on a field K) is discrete if there exists a real number epsilon>0 such that

 |x| in (1-epsilon,1+epsilon)=>|x|=1 for all x in K.
(5)

The p-adic valuation on Q is discrete, but the ordinary absolute valuation is not.

If |·| is a valuation on K, then it induces a metric

 d(x,y)=|x-y|
(6)

on K, which in turn induces a topology on K. If |·| satisfies (4b), then the metric is an ultrametric. We say that (K,|·|) is a complete valuated field if the metric space is complete.


See also

Absolute Value, Local Field, Metric Space, p-adic Number, Strassman's Theorem, Ultrametric, Valuation Group

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References

Cassels, J. W. S. Local Fields. Cambridge, England: Cambridge University Press, 1986.Koch, H. "Valuations." Ch. 4 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 103-139, 2000.Ostrowski, A. "Untersuchungen zur aritmetischen Theorie der Körper." Math. Zeit. 39, 269-404, 1935.van der Waerden, B. L. Algebra, 2 vols. New York: Springer-Verlag, 1991.Weiss, E. Algebraic Number Theory. New York: Dover, 1998.

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Valuation

Cite this as:

Weisstein, Eric W. "Valuation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Valuation.html

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