alternative definition of valuation

One may use a slightly different version of the third criterion to define a slightly more general definition of valuation.

A valuation on a field $𝕂$ is a map $|\cdot |:𝕂\to ℝ$ such that

1. $|x|=0$ if an only if $x=0$

2. $|xy|=|x||y|$

3. $|x+y|\le C\max \{|x|,|y|\}$

The quantity $C$ which appears in the third criterion is a positive real number which is known as the .

There is some flexibility in the choice of the constant $C$ in this definition — one can replace $C$ by a larger number $C^{\prime }$. To deal with this ambiguity, one defines the of the valuation as

The relation of this definition to the usual one is the following. On the one hand, if $|\cdot |$ satisfies the usual definition, then

$$|x+y|\le |x|+|y|\le 2\max \{|x|,|y|\}$$

so a valuation in the old sense is a valuation in the new sense with constant 2.

On the other hand, suppose that $|\cdot |$ satisfies the alternative definition with constant . Then we have the following result.

Theorem If $|\cdot |$ is a valuation according to the definition of this entry with constant $C\le 2$, then $|\cdot |$ satisfies the identity

$$|x+y|\le |x|+|y|.$$

The proof of this assertion is given in a supplement to this entry.

The foregoing discussion shows that the new definition is more general than the old definition precisely when $C>2$. However, this extra generalty is not as great as it might seem at first sight. As is obvious from examining the definition, if $|\cdot |$ is a valuation, then so is $|\cdot {|}^p$ for any power $p>0$. Furthermore, if the valuation has constant $C$, then valuation $|\cdot {|}^p$ has constant $C^p$. Therefore, given any valuation $|\cdot |$ in the sense of this entry, there will exist a number $p$ such that $|\cdot {|}^p$ is a valuation in the sense of the parent entry. Moreover, given the fact that two valuations which are powers of each other are equivalent, one sees that the extra generality is not that interesting since the new valuations are equivalent to the old valuations.