We define two important properties of valuations, both of which
apply to equivalence classes of valuations (i.e., the property
if and only if it holds for a valuation
To say that
is discrete is the same as saying
that the set
if there is a
such that for any
Thus the absolute values are bounded away from
forms a discrete subgroup of the reals under addition (because
the elements of the group are bounded away from 0).
By Proposition 13.2.2, the set
is free on one generator, so there
is a such that
, for ,
runs precisely through the set
is discrete there is a positive
such that for any positive
is an arbitrary positive element.
By subtracting off integer multiples of
find that there is a unique
, it follows
is a multiple of
(Note: we can replace by to see that we
can assume that ).
Axiom (2) of valuations
, we call
Note that if we can take for
then we can take for any valuation equivalent to
To see that (13.2.1) is equivalent to Axiom (3) with
Axiom (3) asserts that
, which implies
, and conversely.
if we can take
in Axiom (3), i.e., if
is not non-archimedean then
it is archimedean
We note at once the following consequence:
is a non-archimedean valuation.
If with , then
is true even if
where for the last equality we have used that
(Ring of Integers)
is a non-archimedean absolute
value on a field
is a ring called the ring of integers
with respect to
Two non-archimedean valuations
are equivalent if and only if they
give the same .
We will prove this modulo the claim (to
be proved later in Section 14.1) that
valuations are equivalent if (and only if) they induce the
is equivalent to
if and only if
, i.e., if
if and only if
The topology induced by
has as basis
of open neighborhoods the set of open balls
, and likewise for
the absolute values
get arbitrarily close
, the set
of open balls
forms a basis of the topology induced
(and similarly for
) we have
so the two topologies both have
a basis, hence are equal. That equal topologies
imply equivalence of the corresponding valuations
will be proved in Section 14.1
The set of with forms an ideal
in . The
is maximal, since if and
, hence , so is a unit.
A non-archimedean valuation
discrete if and only if
is a principal ideal.
First suppose that
we can do since
so the discrete set
is bounded above.
is principal. For any
we have with . Thus
is bounded away from
which is exactly the definition of discrete.
For any prime
, define the
as follows. Write a nonzero
This valuation is both discrete and non-archimedean.
is the local ring
which has maximal ideal generated by
. Note that
We will using the following lemma later (e.g., in
the proof of Corollary 14.2.4 and Theorem 13.3.2).
is non-archimedean if and only if for all in the ring generated by in .
Note that we cannot identify the ring generated by with
in general, because might have characteristic .
is non-archimedean, then
so by Axiom (3) with
, we have
induction it follows that
Conversely, suppose for all integer multiples of .
This condition is also true if we replace
any equivalent valuation, so replace
one with , so that the triangle inequality holds.
Suppose with . Then
by the triangle inequality,
th roots of both sides to get
and take the limit as
. This proves that one
in Axiom (3), hence that