Valuation

A generalization of the p-adic Number first proposed by Kürschák in 1913. A valuation $\vert\cdot\vert$ on a Field $K$ is a Function from $K$ to the Real Numbers $\Bbb{R}$ such that the following properties hold for all $x,y \in K$:

1. $\vert x\vert \geq 0$,

2. $\vert x\vert = 0$ Iff $x=0$,

3. $\vert xy\vert = \vert x\vert\,\vert y\vert$,

4. $\vert x\vert \leq 1$ Implies $\vert 1+x\vert \leq C$ for some constant $C \geq 1$ (independent of $x$).

If (4) is satisfied for $C=2$, then $\vert\cdot\vert$ satisfies the Triangle Inequality,

4a. $\vert x+y\vert \leq \vert x\vert + \vert y\vert$ for all $x,y \in K$.

If (4) is satisfied for $C=1$ then $\vert\cdot\vert$ satisfies the stronger Triangle Inequality

4b. $\vert x+y\vert \leq \max (\vert x\vert, \vert y\vert)$.

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 $\vert\cdot\vert _1$ is a valuation on $K$ and $\lambda \geq 1$, then we can define a new valuation $\vert\cdot\vert _2$ by

$$\vert x\vert _2 = \vert x\vert _1^\lambda.$$ (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. $\Bbb{Q}$, $\Bbb{R}$, and $\Bbb{C}$ with the usual Euclidean norms are Archimedean valuated fields. For any Prime $p$, the p-adic Number $\Bbb{Q}_p$ with the $p$-adic valuation $\vert\cdot\vert _p$ is a non-Archimedean valuated field.

If $K$ is any Field, we can define the trivial valuation on $K$ by $\vert x\vert = 1$ for all $x \not= 0$ and $\vert\vert = 0$, 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 $\Bbb{Q}$ is equivalent to one of the following: the trivial valuation, Euclidean absolute norm $\vert\cdot\vert$, or $p$-adic valuation $\vert\cdot\vert _p$.

The equivalence of any nontrivial valuation of $\Bbb{Q}$ 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 $\vert\cdot\vert _1$, $\vert\cdot\vert _2$, ..., $\vert\cdot\vert _k$ 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

$$\lim_{n\to\infty{\rm\ w.r.t.\ \vert\cdot\vert _1}} x_n = a_1$$ (2)

$$\lim_{n\to\infty{\rm\ w.r.t.\ \vert\cdot\vert _2}} x_n = a_2,$$ (3)

etc. This says that inequivalent valuations are, in some sense, completely independent of each other. For example, consider the rationals $\Bbb{Q}$ with the 3-adic and 5-adic valuations $\vert\cdot\vert _3$ and $\vert\cdot\vert _5$, and consider the sequence of numbers given by

$$x_n = {43\cdot 5^n + 92\cdot 3^n\over 3^n + 5^n}.$$ (4)

Then $x_n\to 43$ as $n\to\infty$ with respect to $\vert\cdot\vert _3$, but $x_n\to 92$ as $n\to\infty$ with respect to $\vert\cdot\vert _5$, 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 $\Bbb{R}$. Equivalently, a valuation (on a Field $K$) is discrete if there exists a Real Number $\epsilon > 0$ such that

$$\vert x\vert \in (1-\epsilon, 1+\epsilon) \Rightarrow \vert x\vert=1 {\rm\ for\ all\ } x \in K.$$ (5)

The $p$-adic valuation on $\Bbb{Q}$ is discrete, but the ordinary absolute valuation is not.

If $\vert\cdot\vert$ is a valuation on $K$, then it induces a metric

$$d(x,y) = \vert x-y\vert$$ (6)

on $K$, which in turn induces a Topology on $K$. If $\vert\cdot\vert$ satisfies (4b) then the metric is an Ultrametric. We say that $(K, \vert\cdot\vert)$ 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

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.