The trivial valuation is the valuation for which
for all . We will often tacitly
exclude the trivial valuation from consideration.
From (2) we have
If and , then by (2).
In particular, the only valuation of a finite field
is the trivial one. The same argument shows that ,
Note that if
is a valuation, then
is also a valuation.
Also, equivalence of valuations is an equivalence relation.
same field are equivalent
if there exists
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
. Thus every valuation is
equivalent to a valuation with . Note that if , e.g.,
is the trivial valuation, then we could
simply take in Axiom (3).
Note that Axioms (1), (2) and Equation (13.1.1) imply Axiom (3)
with . We take Axiom (3) instead of Equation (13.1.1) for
the technical reason that we will want to call the square of the
absolute value of the complex numbers a valuation.
. By Axiom (3) we have
we see that
Also we have
and inductively we have for any
is any positive integer, let
. In particular,
the binomial expansion, we have for any
th roots of both sides to obtain
We have by elementary calculus that
(The ``elementary calculus'': We instead prove that
the argument is the same and the notation is simpler. First, for any
, since upon taking
this is equivalent to
, which is true by hypothesis.
Second, suppose there is an
. Then taking logs of boths sides we see
, so there is no such
. The same argument
swapped implies that
, which proves the lemma.