- (1)
- if and only if ,
- (2)
- , and
- (3)
- there is a constant such that whenever .

The * is the valuation for which
for all . We will often tacitly
exclude the trivial valuation from consideration.
*

From (2) we have

all

If is a valuation and is the constant from Axiom (3), then there is a such that (i.e., ). Then we can take as constant for the equivalent valuation . Thus every valuation is equivalent to a valuation with . Note that if , e.g., if is the trivial valuation, then we could simply take in Axiom (3).

Applying (15.1.2) to and using the binomial expansion, we have for any that

Now take th roots of both sides to obtain

William Stein 2004-05-06